Paper Title: On the spouse-loving variant of the Oberwolfach problem.


Ge-based Clinopyroxene series: first principles and  experimental local probe study
    João P. Araújo
    August 5, 2024
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17pt

§ ABSTRACT 
We prove that K_n+I, the complete graph of even order with a 1-factor duplicated, admits a decomposition into 2-factors, each a disjoint union of cycles of length m ≥ 5 if and only if m|n, except possibly when m is odd and n=4m. In addition, we
show that K_n+I admits a decomposition into 2-factors, each a disjoint union of cycles of lengths m_1,…,m_t, whenever m_1,…,m_t are all even.

Keywords: Oberwolfach Problem, 2-factorization, complete graph plus a 1-factor, resolvable minimum covering by cycles, Spouse-Loving Variant.




§ INTRODUCTION


The well-known Oberwolfach Problem asks the following: Given  t round tables of sizes m_1,…,m_t such that m_1+…+m_t=n, is it possible to seat n people around the t tables for an appropriate number of meals so that every person sits next to every other person exactly once? In graph-theoretic terms, the question is asking whether  K_n can be decomposed into 2-factors, each  a disjoint union of cycles of lengths m_1,…,m_t, whenever m_1+…+m_t=n. Since a graph with odd-degree vertices cannot admit a 2-factorization, Huang, Kotzig, and Rosa <cit.>  proposed the analogous problem for K_n-I, the complete graph of even order with a 1-factor removed. They  called it the Spouse-Avoiding Variant since it models a sitting arrangement of n/2 couples, where each person gets to sit next to every other person, except their spouse, exactly once, and never gets to sit next to their spouse.

For both of these two basic variants of the Oberwolfach Problem,  the cases with uniform cycle length were completely solved decades ago <cit.>. In addition, many solutions are now known for  variable cycle lengths; most notably, the problem is solved for m_1,…,m_t all even <cit.>; for t=2 <cit.>; and for n ≤ 40 <cit.>. In general, however, it is still open.

The Spouse-Avoiding Variant of the Oberwolfach Problem can also be viewed as the maximum packing  variant. This paper, however,  pertains to the minimum covering version of the problem; in other words, we are interested in decomposing K_n+I (the complete graph of even order with a 1-factor duplicated) into 2-factors, each  a disjoint union of cycles of lengths m_1,…,m_t, where m_1+…+m_t=n. We denote this problem by OP^+(m_1,…,m_t), or OP^+(n;m) when m_1=…=m_t=m.
This variant, nicknamed the Spouse-Loving Variant, models a situation where we wish for each person to sit next to exactly one other person — their spouse — twice, and next to every other person exactly once.

Necessary and sufficient conditions for OP^+(n;3) to have a solution were previously determined by Assaf, Mendelsohn, and Stinson <cit.>, and Lamken and Mills <cit.>
under the term resolvable minimum coverings by triples.

<cit.>
OP^+(3t;3) has a solution if and only if t is even and t ≥ 6.


In this paper, we prove the following main result.


Let n be an even integer and m ≥ 5 an integer such that n  4m when m is odd. Then OP^+(n;m) has a solution if and only if m|n.


In addition, we show that the complete solution to the Oberwolfach Problem for bipartite 2-factors <cit.> implies the following.


Let m_1,…,m_t be even integers greater than 3. Then OP^+(m_1,…,m_t) has a solution.




§ PRELIMINARIES


As usual, K_n denotes the complete graph with n vertices, and for n even, K_n-I and K_n+I denote the complete graph K_n with the edges of the 1-factor I deleted and duplicated, respectively. By K_α[k] we denote the complete equipartite graph with α parts of size k. A cycle of length m (or m-cycle) is denoted by C_m.

A set { H_1,…, H_k } of subgraphs of a graph G is called a decomposition of G if { E(H_1),…, E(H_k) } is a partition of E(G). If this is the case, we write G=H_1 ⊕⋯⊕ H_k.

A 2-factor in a graph G is a spanning 2-regular subgraph of G. A 2-factor consisting of disjoint cycles of lengths m_1, m_2,…, m_t, respectively, is called a (C_m_1,C_m_2,…,C_m_t)-factor, and a
(C_m,C_m,…,C_m)-factor is also called a C_m-factor.

A 2-factorization of a graph G is a decomposition of G into 2-factors. A (C_m_1,C_m_2,…,C_m_t)-factorization is a  2-factorization consisting of (C_m_1,C_m_2,…,C_m_t)-factors; a C_m-factorization is defined analogously.

Apart from direct constructions, the following two previous results will form the most important tools for the proofs of our two main theorems.

<cit.>
Let α, k, and ℓ be positive integers, α≥ 2, and 3 ≤ℓ≤α k. Then K_α[k] admits a C_ℓ-factorization if and only if


  * α k ≡ 0 ℓ,

  * k(α-1) is even,

  * ℓ is even if α=2, and

  * (k,α,ℓ) ∉{ (2,3,3),(6,3,3),(2,6,3),(6,2,6) }.



<cit.> 
Let m_1,…,m_t be even integers greater than 3. Then OP(m_1,…,m_t) has a solution, that is, K_n-I admits a (C_m_1,C_m_2,…,C_m_t)-factorization for n=m_1+… +m_t.





§ RESULTS


Our Theorem <ref> is an easy corollary to Theorem <ref> <cit.>.



 <ref>.
Let n=m_1+… +m_t, and let I denote a chosen 1-factor of K_n. By Theorem <ref> <cit.>, there exists a (C_m_1,C_m_2,…,C_m_t)-factorization of K_n-I. It is easy to see that, since all m_i are even, there exists a 1-factor I' of K_n such that I ⊕ I' is a (C_m_1,C_m_2,…,C_m_t)-factor. Thus K_n ⊕ I' admits a (C_m_1,C_m_2,…,C_m_t)-factorization, and OP^+(m_1,…,m_t) has a solution.


We now turn our attention to odd-length cycles.






For all odd m ≥ 5, there exists a C_m-factorization of K_2m+I.



Let the vertex set of K_2m+I be
V={ x_i: i ∈_m }∪{ y_i: i ∈_m }.
Edges of the form x_i x_i+d and y_i y_i+d will be called of pure left and pure right difference d, respectively, while an edge of the form x_i y_i+d will be called of mixed difference d.

Define a permutation ρ=(x_0   x_1… x_m-1) (y_0   y_1… y_m-1). In each of the cases below, we construct a starter C_m-factor F containing exactly one edge of each pure left and pure right difference,  exactly two edges of a single mixed difference, and exactly one edge of every other mixed difference. It then easily follows that {ρ^ℓ(F): ℓ∈_m} is a C_m-factorization of K_2m+I.
Figure <ref> shows the starter 2-factors in a C_5-factorization of K_10+I and a C_7-factorization of K_14+I, respectively.





Case m ≡ 3  4.


First assume m  11. Define the following walks (Figure <ref>).

    P    =    x_0 y_0 x_1 y_-1 x_2 y_-2… x_m-3/4 y_-m-3/4 x_m+5/4, 
    
    P'    =    y_1 x_-1 y_2 x_-2… y_m-3/4 x_-m-3/4 y_m+5/4 x_m+1/4 y_m+1/4,  
    
    Q    =    x_m+5/4 x_-m+1/4 x_m+9/4 x_-m+5/4… x_m-1/2 x_-m-3/2 x_-m-1/2 x_0.

It is not difficult to verify that each of these walks is in fact a path, and that these three paths are pairwise vertex-disjoint, except that P and Q have the same pair of endpoints.

Observe that P contains exactly one edge of each of the mixed differences in

    D(P)={ 0,-1,-2,-3,…,-m-3/2, m-1/2},

P' contains exactly one edge of each of the mixed differences in

    D(P')={ 0,1, 2,3,…,m-3/2, -m-1/2},

while Q contains exactly one edge of each pure left difference in

    D(Q)={ 1,2,…, m-1/2}.


Our starter 2-factor will contain the m-cycle C=PQ, as well as the m-cycle C'=P'Q', where Q' is the (y_m+1/4,y_1)-path to be defined as follows below.

First, for i ≤m-9/2 of the form i=m+1/4+3j, where j ∈, define a 6-path

    Q_i= y_i y_-(i+1) y_i+2 y_-i y_i+4 y_-(i+2) y_i+3.

Observe that Q_i contains exactly one edge of each of the pure right differences in

    D(Q_i)={ -(2i+1), -(2i+2),…,-(2i+6) }.

Thus, if j>0, paths Q_i and Q_i+3j have no edges of the same difference, and share no vertices except that the terminus of Q_i  equals the
source of Q_i+3. We use paths Q_i to define Q' as follows.

If m ≡ 3 12, let

    Q'= Q_m+1/4 Q_m+13/4 Q_m+25/4… Q_m-13/2 y_m-7/2 y_-m-5/2 y_m-3/2 y_-m-7/2 y_-m-1/2 y_m-1/2  y_-m-3/2 y_1.




If m ≡ 7 12, let

    Q'= Q_m+1/4 Q_m+13/4 Q_m+25/4… Q_m-9/2 y_m-3/2 y_-m-1/2 y_-m-3/2  y_1.



If m ≡ 11 12, let

    Q'= Q_m+1/4 Q_m+13/4… Q_m-17/2 y_m-11/2 y_-m-9/2 y_m-7/2 y_-m-11/2 y_m-3/2 y_-m-5/2 y_m-5/2 y_-m-7/2
    y_-m-1/2 y_m-1/2  y_-m-3/2 y_1.

Note that for m ∈{ 7,15,23}, the walk Q' contains no paths Q_i.

It is not difficult to verify that in each case, Q' is in fact a path, and contains exactly one edge of each pure right difference in

    D(Q')={ 1,2,…,m-1/2}.


Moreover, the paths P' and Q' are internally vertex-disjoint and share the endpoints, so C'=P'Q' is an m-cycle. Furthermore, the m-cycles C and C' are disjoint, and F={ C,C'} is an C_m-factor in K_2m+I containing exactly one edge of each pure left difference, each pure right difference, and each mixed difference, except that it contains exactly two edges of mixed difference 0. Hence {ρ^ℓ(F): ℓ∈_m} is a C_m-factorization of K_2m+I.

For m=11, we define P and Q as above, and modify P' and Q' as follows:

    P'    =    y_1 x_-1 y_2 x_-2 y_4 x_3 y_5, 
    
    Q'    =    y_5 y_-4 y_-5 y_3 y_-3 y_1.

We obtain a C_m-factorization of K_2m+I (for m=11) as before. Note that, in this case, the duplicated mixed difference is 2.






Case m ≡ 1 4. 
First assume m ≥ 13. Define the following walks (Figure <ref>).

    P    =    x_0 y_0 x_1 y_-1 x_2 y_-2… x_m-1/4 y_-m-1/4 x_-m+3/4, 
    
    P'    =     y_1 x_-1 y_2 x_-2…  y_m-1/4 x_-m-1/4 x_m+7/4 y_m+3/4, 
    
    Q'    =    y_m+3/4 y_-m+3/4 y_m+7/4 y_-m+7/4… y_m-3/2 y_-m-3/2  y_m-1/2 y_-m-1/2 y_1.


It is not difficult to verify that each of these walks is in fact a path, and that these three paths are pairwise vertex-disjoint, except that P' and Q' have the same endpoints.

Observe that P contains exactly one edge of each of the mixed differences in

    D(P)={ 0,-1,-2,…,-m-1/2, 1 },

P' contains an edge of pure left difference m-3/2 and exactly one edge of each of the mixed differences in

    D(P')={ -1,2,3,…,m-1/2},

and Q contains exactly one edge of each pure right difference in

    D(Q')={ 1,2,…, m-1/2}.


Next, let C'=Q'P' and observe that C' is an m-cycle. The second cycle in our starter 2-factor will be C=PQ, where Q is the (x_-m+3/4,x_0)-path to be defined as follows below.

First, for i ≤m-11/2 of the form i=m+19/4+3j, where j ∈,  define a 6-path

    Q_i= x_-(i-3) x_i+2 x_-(i+2) x_i+1 x_-(i+1) x_i x_-i .

Observe that Q_i contains exactly one edge of each pure left difference in

    D(Q_i)={ -(2i-1), -2i,…,-(2i+4) }.

Thus, for j>0, paths Q_i and Q_i+3j  contain no edges of a common difference, and share no vertices except that the terminus of Q_i equals the source of Q_i+3.

For m ≥ 17, we additionally  define

    Q_m+7/4= x_-m+3/4 x_m+15/4
    x_-m+15/4 x_m+11/4
    x_-m+11/4 x_m+3/4
    x_-m+7/4.

Thus Q_m+7/4 contains exactly one edge of each pure left difference in

    D(Q_m+7/4)={m-15/2,m-13/2,…,m-5/2}

and shares no differences and no vertices with any path Q_m+19/4+3j for j ∈, except that the terminus of Q_m+7/4 equals the source of Q_m+19/4. We use the paths Q_i to define Q as follows.

If m ≡ 1 12 and m ≥ 37, let

    Q= Q_m+7/4 Q_m+19/4… Q_m-15/2
    x_-m-15/2 x_m-5/2 x_m-1/2 x_-m-9/2 x_m-9/2 x_-m-7/2
    x_m-7/2 x_-m-5/2
    x_m-3/2 x_-m-3/2
    x_-m-1/2 x_0.

Modifications are required for small values of m: for m=13, we take Q=x_-4 x_6 x_4 x_-5 x_-6 x_0, and
for m=25, we  take
Q=x_-7 x_10 x_12 x_-8 x_7 x_-9 x_9 x_-10 x_11 x_-11 x_-12 x_0 instead.

If m ≡ 5 12 and m ≥ 29, let

    Q= Q_m+7/4 Q_m+19/4 Q_m+31/4… Q_m-11/2
    x_-m-11/2 x_m-1/2 x_m-3/2 x_-m-3/2 x_m-5/2 x_-m-5/2 x_-m-1/2 x_0,

while for m=17, we take
Q=x_-5 x_8 x_7 x_-7 x_5 x_-6 x_-8 x_0.

If m ≡ 9 12 and m ≥ 33, let

    Q= Q_m+7/4 Q_m+19/4 Q_m+31/4… Q_m-13/2
    x_-m-13/2 x_m-3/2 x_m-1/2 x_-m-3/2 x_m-5/2 x_-m-5/2
    x_m-7/2 x_-m-7/2
    x_-m-1/2 x_0,

while for m=21, take
Q=x_-6 x_9 x_10 x_-9 x_8 x_-8 x_6 x_-7 x_-10 x_0 instead.

It is not difficult to verify that in each case, the walk Q is in fact a path, and contains exactly one edge of each pure left difference in

    D(Q)={ 1,2,3, …,m-5/2,m-1/2}.

Recall that pure left difference m-3/2 was used in the cycle C'.

Since paths P and Q are internally vertex-disjoint and share their endpoints, C=PQ is indeed an m-cycle. Moreover, F={ C,C'} is a pair of disjoint m-cycles in K_2m+I containing exactly one edge of each pure left difference, each pure right difference, and each mixed difference, except it contains exactly two edges of mixed difference -1. Hence F is a starter 2-factor of a C_m-factorization of K_2m+I.

For m=9, we define paths P and Q' as before, and  modify P' and Q as

    P' = y_1 x_-1 y_2 x_-2 x_-4 y_3          Q=x_-3 x_3 x_4 x_0.

In this case, the duplicated mixed difference is -2.

Finally, for m=5, we let

    C=x_0 y_1 y_-1 x_-1 x_-2x_0           C'=y_0 x_1 y_-2 y_2 x_2 y_0,

so that F={ C,C'} is a starter 2-factor with a duplicated mixed difference 0.

This completes all cases of the construction of a C_m-factorization of K_2m+I.




We round up this section with the proof of our main result.




 <ref>.
Let n be an even integer and m ≥ 5. If OP^+(n;m) has a solution, then clearly m|n.

Conversely, assume n is even and n=tm for an integer t. If m is even, then OP^+(n;m) has a solution by Theorem <ref>. Hence assume m is odd and t  4. The result follows from Lemma <ref> if t=2. Hence let t ≥ 6. Decompose K_tm+I into t/2 disjoint copies of K_2m+I and the complete equipartite graph with t/2 parts of size 2m. The first graph admits a C_m-factorization by Lemma <ref>, and the second by Theorem <ref>. Hence K_n+I admits a C_m-factorization, and OP^+(n;m) has a solution.




§ CONCLUSION


The main result of this paper is an almost complete solution to the Spouse-Loving Variant of the Oberwolfach Problem with uniform cycle lengths. The only case that remains unsolved is captured by the following conjecture.


Let m be a positive odd integer. Then OP^+(4m;m) has a solution if and only if m ≥ 5.


It is worthwhile observing that this case is analogous to the only case left unsolved in <cit.>, where the Oberwolfach Problem with uniform cycle lengths was almost completely solved. The reason is the same: the reduction to case n=2m does not work since K_2m,2m admits no C_m-factorization when m is odd. A solution to OP(4m;m), which appeared in <cit.> two years after <cit.>, required a rather different and highly involved construction.


The reader may be wondering whether a solution to OP^+(4m;m) may be constructed similarly to our solution to OP^+(2m;m). Suppose this is indeed the case. Analogously to the proof of  Lemma <ref>, denote V(K_4m+I)= { x_i: i ∈_2m}∪{ y_i: i ∈_2m} and ρ=(x_0   x_1 … x_2m-1)(y_0   y_1 … y_2m-1), and let F be a starter 2-factor such that {ρ^i(F): i ∈_2m} is a C_m-factorization of K_4m+I. Then F contains exactly one edge of each difference in K_4m+I, and since the edges of pure left difference m and pure right difference m lie in orbits of length m, they must be the ones forming the duplicated 1-factor I. Hence the freedom of choosing I that we enjoyed in the case n=2m is lost. However, the fatal flaw is as follows. The graph K_4m+I contains edges of exactly 4m distinct differences (which is promising), of which exactly 2m+1 are odd (m+1/2 of the pure left differences, m+1/2 of the pure right differences, and m of the mixed differences). However, each cycle in F must contain an even number of odd differences — a contradiction.

Thus, proving Conjecture <ref> will require a different approach. So far we have been able to verify it for 5 ≤ m ≤ 23, using a combination of general construction, case-specific construction, and, for larger m, a computer search. However, despite considerable effort, we have to conclude that a complete proof of Conjecture <ref> is presently out of our reach.



Acknowledgement



A. Burgess and M. Šajna gratefully acknowledge support by the Natural Sciences and Engineering Research Council of Canada.


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 Paper Title: Representability of the direct sum of uniform q-matroids.





§ ABSTRACT 
  There are many similarities between the theories of matroids and q-matroids. However, when dealing with the direct sum of q-matroids many differences arise. Most notably, it has recently been shown that the direct sum of representable q-matroids is not necessarily representable. In this work, we focus on the direct sum of uniform q-matroids. Using algebraic and geometric tools, together with the notion of cyclic flats of q-matroids, we show that this is always representable, by providing a representation over a sufficiently large field.


Driving of an open quantum system at finite temperature across first- and second-order quantum phase transitions
    Pavel Cejnar
    August 5, 2024
================================================================================================================



Keywords. q-matroids, representability, evasive subspaces, rank-metric codes, linear sets.










§ INTRODUCTION


The concept of q-matroid traces back to Crapo's PhD thesis <cit.>. More recently, q-matroids have been reintroduced in <cit.> and they gained a lot of attention because of their relation to rank-metric codes; see for instance <cit.>.  

A full exposition of cryptomorphisms, i.e. many equivalent ways to describe a q-matroid axiomatically, has been given in <cit.>, in terms of rank function, independent spaces, flats, circuits, bases, spanning spaces, closure function, hyperplanes, open spaces etc. Later, another cryptomorphism based on cyclic flats has been derived in <cit.>. In general, these are not straightforward q-analogues of the traditional matroids' ones. 

While there are many similarities between matroids and q-matroids, there are substantial differences when dealing with the direct sum. Recently, the direct sum of q-matroids has been introduced in terms of the rank function; see <cit.>. In the same paper, some fundamental properties have been established. Moreover in <cit.>, it has been shown that the direct sum of q-matroids is a coproduct in the category of q-matroids with linear weak maps as morphisms. This also implies that the direct sum of q-matroids on ground spaces E_1 and E_2 is the unique q-matroid with the most independent spaces among all q-matroids on E_1 ⊕ E_2 whose restrictions to E_i are isomorphic to the given q-matroids. In <cit.>, it is shown that the cyclic flats are one of the few objects that behave well with the direct sum of q-matroids. In <cit.>, the first steps towards the study of the representation of the direct sum of q-matroids have been taken. For a q-matroid with ground space _q^n, representability can be defined over any field extension of _q; see Section <ref>. One big difference with the representation of matroids is that in q-matroids the characteristic of the field is fixed, and the only free parameter is the degree of the field extension. Moreover, while the direct sum of matroids is always representable, it has been shown that the direct sum of two representable q-matroids may not be representable over any field, or it may require a sufficiently large field; see <cit.>. Representability of q-matroids has been geometrically described in <cit.>, where it is illustrated how to equivalently define the rank function of a representable q-matroid in terms of a q-system, a notion introduced in <cit.>.






  
Our contribution: In this paper we study the direct sum of t many uniform q-matroids. These are special q-matroids, since they are representable as  maximum rank distance (MRD) codes or (equivalently) as scattered subspaces.

Our first main result is the geometric characterization of the representation of the direct sum of uniform q-matroids. We prove that a q-system is a representation of such direct sum if and only if it satisfies some evasiveness properties; see Theorem <ref>. 
The crucial objects towards our characterization are the cyclic flats, which for uniform q-matroids are just the 0-subspace and the whole ground space. 

The second main result is the construction of a q-system (over a sufficiently large field) that satisfies the previously mentioned evasiveness properties; see Theorem <ref>. This, in particular, shows that the direct sum of uniform q-matroids is always representable; see Theorem <ref>. However, it remains  an open question which is the smallest extension field over which we can find a representation of this direct sum. 

In the last part of the paper we study the extension fields over which the direct sum of two uniform q-matroids of rank 1 is representable. This is a special case, since the associated system has been already investigated in terms
of linear sets with complementary weights; see <cit.>. Thanks to this point of view, we can provide more precise results on the field size needed for the representation of two uniform q-matroids of rank 1; see Theorem <ref>.






  
Outline of the paper: The rest of the paper is organized as follows. In Section <ref>, we provide the needed background. In Section <ref>, we characterize the direct sum of t many uniform q-matroids in geometric terms. In Section <ref>, we exhibit a representation of such direct sum. In Section <ref>, we restrict ourselves to study the direct sum of two uniform q-matroids of rank 1. We draw some conclusions and open problems in Section <ref>.





§ ACKNOWLEDGMENTS

Gianira N. Alfarano is supported by the Swiss National Foundation through grant no. 210966 and by the FWO (Research Foundation Flanders) grant no. 1273624N. Alessandro Neri is supported by   the FWO (Research Foundation Flanders) grant no. 12ZZB23N. Ferdinando Zullo is very grateful for the hospitality of the Ghent University, where he was a visiting researcher during the development of this research in September 2023. He is supported by the project COMBINE of “VALERE: VAnviteLli pEr la RicErca" of the University of Campania “Luigi Vanvitelli” and by Erasmus+ UE (Programme Countries) from Unicampania.
This research is partially supported by the Italian National Group for Algebraic and Geometric Structures and their Applications (GNSAGA - INdAM).






§ PRELIMINARIES


In this section we provide the necessary background for the rest of the paper. 




 §.§ Rank-metric codes and q-systems


Rank-metric codes have been introduced originally by Delsarte in <cit.> and by Gabidulin in <cit.>. Let _q be the finite field with q elements and let m,n be positive integers with m≥ n. We endow the vector space _q^m^n with the rank metric, defined for every x=(x_1,…,x_n), y=(y_1,…,y_n)∈_q^m^n as 

    (x,y) = __q⟨ x_1-y_1,…, x_n-y_n⟩__q.

 An [n,k]_q^m/q rank-metric code  is a k-dimensional _q^m-linear subspace of _q^m^n. The minimum rank distance of  is defined as
 
    ():=    min{(x,y) | x,y ∈,   x≠ y }.


If d=() is known, we write that  is an [n,k,d]_q^m/q  code.
The parameters n,m,k,d of an [n,k,d]_q^m/q  rank-metric code satisfy a Singleton-like bound <cit.>, that reads as

    k ≤ n-d+1.

When equality holds, we say that  is a maximum rank distance code or MRD for short.
It has been shown that MRD codes exist if and only if n≤ m, for any choice of d≤ n, over any finite field, and over any field admitting a cyclic Galois extension of degree degree m; see <cit.>. Note that we are defining MRD codes starting from the Singleton-like bound in (<ref>). However, sometimes in the literature one also admits the transpose Singleton-like bound, for which MRD codes may exist also when n>m.

For a vector 
v ∈_q^m^n and an ordered basis Γ={γ_1,…,γ_m} of the field extension
_q^m/_q, let Γ(v) ∈_q^n × m be the matrix defined by

    v_i= ∑_j=1^m Γ(v)_ijγ_j.

The support of v is the column space of Γ(v) for any basis Γ; we denote it by (v).
The rank-weight of v∈_q^m^n is the quantity (v)=__q((v)).
A generator matrix for an [n,k,d]_q^m/q  code  is a full-rank matrix G∈_q^m^k× n such that =(G). If the columns of one (and hence any) generator matrix of  are _q-linearly independent, then  is said to be nondegenerate. 
Two [n,k,d]_q^m/q rank-metric codes  and ' are equivalent if and only if
there exists A ∈GL(n,_q) such that
'= A={vA : v ∈}. 


There is a geometric interpretation of [n,k,d]_q^m/q rank-metric codes as q-systems. 
 
An  system  is an n-dimensional _q-subspace of _q^m^k, such that
⟨⟩__q^m=_q^m^k and

    d=n-max{__q(∩ H) | H  is an _q^m-hyperplane of _q^m^k}.

Moreover, two [n,k,d]_q^m/q systems  and ' are equivalent if there exists an _q^m-isomorphism φ∈(k,_q^m) such that

    φ() = '.

If the parameters are not relevant or clear from the context, we simply say that  is a q-system.




Although the term “q-system” has been introduced only a few years ago, the theory of these objects is strictly related to the one of linear sets. In some sense, q-systems and linear sets can be thought of as the same concept. Thus, we will report a few results and properties originally studied in the theory of linear sets, using the language of q-systems. The interested reader is referred to <cit.> for an in-depth treatment of linear sets. 


Let  be an  system. For each -subspace V of  ^k, we define the weight of V in  as the integer

    _(V):=__q(∩ V).




In <cit.> it has been proved that there is a one-to-one  correspondence  between  equivalence  classes of nondegenerate  codes and equivalence classes of  systems and the correspondence can be explained as follows.
Let  be an [n,k,d]_q^m/q code and G be a generator matrix for . Define  to be the _q-span of the columns of G. In this case  is also called a q-system associated with . Vice versa, given an [n,k,d]_q^m/q system , define G to be the matrix whose columns are an _q-basis of  and let  be the code generated by G.  is also called a code associated with . For more details on this correspondence, see also <cit.>.



Finally recall the following result which provides a natural description of the supports of codewords of an [n,k]_q^m/q code  in terms of a q-system  associated to .

Let  be an [n,k]_q^m/q non-degenerate rank-metric code and let  be the _q-span of the columns of a generator matrix G. Consider the isomorphism

    ψ_G: _q^n →,     v ↦ vG^⊤.

For every u∈_q^m^k we have that 

    ψ_G^-1(∩⟨ u ⟩ ^⊥)=(uG)^⊥.



 
Let 𝒜 be a collection of _q-subspaces of _q^m^k, let  be an  system and let h be a positive integer. We say that
 is (𝒜,h)-evasive if

    _(A)≤ h,   .



Remark that since every _q^m-linear subspace is also _q-linear, we can apply the above definition to _q^m-linear subspaces as well.

Denote by Λ_h the set of all the h-dimensional _q^m-subspaces of _q^m^k, that is

    Λ_h:={ V : V  is _q^m-subspace of _q^m^k  and _(V)=h}.



    Let  be an  system, and let h,r be positive integers such that 1≤ h ≤ k. We say that  is (h,r)-evasive if  is (Λ_h,r)-evasive. When h=r, we say that  is h-scattered. Finally, a 1-scattered q-system will be simply called scattered. 



The study of scattered q-systems originated in <cit.>. This notion was then generalized to any h in <cit.>, although the (k-1)-scatteredness was previously used in <cit.> due to its connection with MRD codes. The complete correspondence between maximum h-scattered systems and MRD codes was then highlighted in <cit.>. More recently, the more general property of evasiveness was analyzed in <cit.> and <cit.>. The connection between evasiveness of q-systems and generalized rank weights of a corresponding code can be instead found in <cit.>.


Finally, we introduce the following notion for a q-system. 

    For an -subspace V⊆^k, we define the -rank of V to be the integer
    
    ρ(V)=_(⟨ V⟩_),

    that is the _q^m-dimension of the _q^m-span of any _q-basis of V.
    When we consider ρ restricted to the -subspaces of an  system , we will write ρ_.




Given a vector space E, we denote by (E) the lattice of all subspaces of E.
For any direct sum E=E_1⊕ E_2⊕⋯⊕ E_t of _q-vector spaces E_1, E_2, …, E_t, we denote by π_i:E⟶ E_i  the corresponding projections and by ι_i:E_i⟶ E the canonical embedding.
Furthermore, for n_1,n_2,…, n_t∈ℕ and n=n_1+n_2+⋯+n_t, we set _q^n=_q^n_1⊕_q^n_2⊕⋯⊕_q^n_t such that
π_i:_q^n⟶_q^n_i  is the projections onto the corresponding n_i coordinates. Let Q be a prime power and define ι_i:(𝔽_Q^n_i)⟶(𝔽_Q^n) such that for any X ∈(𝔽_Q^n_i) then 
    ι_i(X)=⟨ 0 ⟩⊕⋯⊕⟨ 0⟩⊕ X⊕⟨ 0 ⟩⊕⋯⊕⟨ 0 ⟩.





    Let _1,…,_t be [n_i,k_i]_q^m/q systems, for 1≤ i≤ t. Then, denote by ⊕_i=1^t_i the [∑_i=1^t n_i, ∑_i=1^t k_i]_q^m/q system given by  ι_1(_1)+…+ι_t(_t). 








 §.§ q-Matroids

In this subsection we briefly recall some facts about q-matroids that will be useful later on.

    A q-matroid with ground space E is a pair =(E,ρ), where E is a finite dimensional _q-vector space and ρ is an integer-valued 
	function defined on (E) with the following properties:
	
		
  (R1) Boundedness: 0≤ρ(A) ≤ A, for all A∈(E). 
		
  (R2) Monotonicity: A≤ B ⇒ρ(A)≤ρ(B), for all A,B∈(E). 
		
  (R3) Submodularity: ρ(A+ B)+ρ(A∩ B)≤ρ(A)+ρ(B), for all A,B∈(E).  
	
 The function ρ is called rank function and the value ρ() := ρ(E) is the rank of the q-matroid. The value h()=__q(E) is the height of .


A 1-dimensional space x∈(E) is a loop if ρ(x)=0. A subspace A∈(E) is independent if ρ(A)=(A) and dependent otherwise. The inclusion-minimal dependent spaces are called circuits. A space A∈(A) is a flat if it is inclusion-maximal in the set {V ∈(E) |ρ(V)=ρ(A)}, and a space is open if it is the sum of circuits. Finally, a subspace A∈ which is both a flat and an open space is called a cyclic flat. 

We denote by () and () the collections of independent spaces and cyclic flats of the q-matroid , respectively.
If it is clear from the context, we will simply write ,.

It is well known that the collection of independent spaces uniquely
determines the q-matroid, and the same is true for the collections of dependent spaces, open spaces,
and circuits. For this and many more cryptomorphisms for q-matroids we refer to <cit.>. Furthermore, the cyclic flats
along with their rank values  also uniquely determine the q-matroid; see <cit.>.

Two q-matroids _i=(E_i,ρ_i), i=1,2 are equivalent if there exists a rank-preserving -isomorphism α:E_1→ E_2, i.e. ρ_1(V)=ρ_2(α(V)) for all V∈(E_1). When this happens, we will write _1≅_2.

We recall a well-known family of q-matroids, namely the family of uniform q-matroids; see for instance <cit.>.


Let 0≤ k≤ n. For each V∈(_q^n), define ρ(V) :=min{k,(V)}. Then (_q^n,ρ) is a q-matroid. It is called the uniform q-matroid on _q^n of rank k and is denoted by _k,n(q).




An important construction of a q-matroid arises from matrices; see <cit.>. Let G be a k × n matrix with entries in _q^m and for every U∈(_q^n), let A^U be a matrix whose columns form a basis of U. Then the map 

    ρ:(_q^n) →ℤ,   U↦(G A^U),

is the rank function of a q-matroid, which we denote by _G and we call the q-matroid represented by G. A q-matroid  with ground space _q^n is called _q^m-representable if =_G for some matrix G with  n columns and entries in _q^m, whose rank equals the rank of . The q-matroid  is called representable if it is _q^m-representable over some field extension _q^m.
In other words, we can say that  is _q^m-representable if there is a rank-metric code with generator matrix G, such that =_G. 

There is also an equivalent geometric interpretation of a representable q-matroid in terms of q-systems. This was proved in <cit.>, whose original statement may look slightly different. However, it is equivalent to the following reformulation.

 
Let G be the generator matrix of a nondegenerate  code, and let  be any  system associated to it. Them _G≅ (,ρ_).


In other words, Theorem <ref> characterizes representable q-matroids as those coming from a q-system. More precisely, a q-matroid  of rank k and height n without loops is -representable if and only if it is equivalent to  a q-matroid (,ρ_), for some  system . 


    In the original statement <cit.>, the equivalence _G≅ (,ρ_) is derived by characterizing the collection of independent spaces (_G). This is  given by the set of -independent, which are precisely the -subspaces I⊆ such that _(I)=_(⟨ I⟩_). Since by definition of ρ_ (see Definition <ref>) the independent spaces of (,ρ_) coincides with (_G), then Theorem <ref> is equivalent to <cit.>.



 
Let _k,n(q) be the uniform q-matroid of rank k with ground space _q^n. Then the trivial uniform q-matroid _0,n(q) and the free uniform q-matroid _n,n are
representable over _q. In particular, _0,n(q) is represented by the 1 × n zero matrix and _n,n(q) by
the n× n identity matrix. For 0 < k < n, the uniform q-matroid _k,n(q) is representable over _q^m if and only if
m ≥ n. Indeed, a matrix G∈_q^m^k× n represents _k,n(q) if and only if (GY^⊤) = k for all Y ∈_q^k× n of rank k.
But this is equivalent to G generating an MRD code and such a matrix G exists
if and only if m≥ n; see e.g. <cit.>. 
Analogously, in the language of q-systems, a -representation of 𝒰_k,n(q) is given by any  system which is (k-1)-scattered, and this is known to exist if and only if m≥ n; see e.g. <cit.>.





Like in the classical case, not all q-matroids are representable; see for instance the Vámos q-matroid defined in <cit.>. However, so far, very little is known about representability of q-matroids. 



 §.§ The direct sum of q-matroids

In <cit.> the notion of direct sum of two q-matroids has been introduced. We are going to define the direct sum of t many q-matroids iteratively, thanks to the associativity property of the direct sum, proved in <cit.>. We use Notation <ref>.


    Let _i=(E_i,ρ_i), i∈[t], be q-matroids and set E=E_1⊕⋯⊕ E_t.
Define ρ'_i:(E)⟶ℕ_0, V⟼ρ_i(π_i(V)) for i∈[t].
Then '_i=(E,ρ'_i) is a q-matroid. 
We define the direct sum of _1, …,_t to be the q-matroid :=(E,ρ), where ρ is defined iteratively as follows:


  * If t=2, then

    ρ:(E)⟶ℕ_0,   V⟼ V+min_X≤ V(ρ'_1(X)+ρ'_2(X)- X).


  * If t>2, then ρ is the rank function of (_1⊕⋯⊕_t-1)⊕_t.




In <cit.>, the authors initiated the study of the representability of the direct sum of two q-matroids. In particular, they proved that if the direct sum is representable then a representation is given by a block diagonal matrix whose blocks represent the
summands. In the following, we list a few important results from <cit.>, reformulated in the language of q-systems.






Let _i=(_q^n_i,ρ_i),i=1,2, be q-matroids of rank k_i, and let  =_1⊕_2.
Suppose  is representable over _q^m.
Then _1 and _2 are representable over _q^m and =(_1⊕_2,ρ__1⊕_2) for some [n_i,k_i]_q^m/q systems _i, i ∈{1,2}.
Furthermore, (_i,ρ__i) is an -representation of _i for each i ∈{1,2}. 




Moreover, in <cit.>, it is shown that the direct sum of two _q^m-representable q-matroids may not be _q^m-representable or, even, not be representable over any field extension of _q. 

[<cit.>]
Let = _1,2(q). Then  is representable over _q^2, whereas ⊕ is representable over _q^m if and only if m ≥ 4.



[<cit.>]
    Let _q=_2 and _4={0,1,ω,ω+1}. Consider the full [4,2]_4/2 system 
=_4^2
and set :=(,ρ_).
Then ⊕ is not representable.




In the rest of the paper we investigate the representability of the direct sum of t uniform q-matroids. To this end, we will use the following properties of cyclic flats of q-matroids.




Let =(E,ρ) be a q-matroid. I∈(E) is independent if and only if for every cyclic flat X∈(E), (I ∩ X) ≤ρ(X).



Let =(_q^n,ρ) be a q-matroid of rank k, with 0 < k < n. Then =_k,n(q) if and only if  has only two cyclic flats, namely ⟨ 0⟩ and _q^n.


In <cit.>, it ha been shown that cyclic flats also behave well with the direct sum of q-matroids. We state the result for t summands.




    For each i∈ [t], let _i=(_q^n_i,ρ_i) be a q-matroid. Then,     
    (_1⊕⋯⊕_t)={⊕_i=1^tZ_i :  Z_i∈(_i),   for  i∈[t]}.








§ THE DIRECT SUM OF UNIFORM Q-MATROIDS


In this section we take the first steps toward the geometric characterization of the representability of the direct sum of t uniform q-matroids _k_1,n_1⊕⋯⊕_k_t,n_t, by characterising its independent spaces. In the rest of the paper we will assume that none of the t summands is the trivial or the free uniform q-matroid. This is due to the following observation.


 
The trivial uniform q-matroid _0,s(q) has only loops, so we cannot consider its representation as q-system. However, as we already pointed out, in terms of matrices,  _0,s(q)  can represented by a 1× s zero matrix. Moreover, it is not difficult to see that a matrix G∈^k× n is an -representation of a q-matroid  if and only if the matrix 

    ( G | 0 )∈^k ×(n+s)

is an -representation of ⊕_0,s(q). 

 Furthermore, in <cit.> it is proven that a q-matroid  is _q^m-representable as =(,ρ_), if and only if ⊕_s,s(q) is _q^m-represented by (⊕^s,ρ_⊕^s). 

Hence, without loss of generality, we can assume from now on that none of the summands involved is the trivial uniform q-matroid nor the free uniform q-matroid. 


For the rest of the paper, assume 1≤ k_i<n_i≤ n and let _k_i,n_i(q), i∈[t] be uniform q-matroids. 

By Lemmas <ref> and <ref>, we have that

    (_k_1,n_1(q)⊕⋯⊕_k_t,n_t(q))={⊕_i=1^t Z_i  :  Z_i∈{⟨ 0 ⟩, _q^n_i}, i ∈ [t]}.


Moreover, it follows from <cit.> that

    ρ(⊕_i=1^t Z_i)=∑_i=1^tρ_i(Z_i),

where Z_i∈{⟨ 0 ⟩, _q^n_i}. 


The next theorem finds the independent spaces of the direct sum of t uniform q-matroids in terms of the cyclic flats of the direct sum.


    Let k_1,…, k_t,n_1,…, n_t,m,r be positive integers, with 1≤ k_i< n_i≤ m for i ∈ [t].
    Let =_k_1,n_1(q)⊕⋯⊕_k_t,n_t(q) and for each 𝒥⊆ [t] denote k_𝒥:=∑_j∈𝒥k_j. Then I∈() if and only if 
    
    (I ∩(∑_j ∈𝒥ι_j(^n_j)))≤ k_𝒥,      𝒥⊆ [t].

    


    It follows directly from Lemmas <ref> and <ref> and the discussion above.


Let 𝐤=(k_1,…,k_t)∈ℕ^t and let h,k be  positive integers such that k=k_1+…+k_t and 1≤ h ≤ k-1, then 
we define 
    Λ_h,𝐤:={ V : V  is _q^m-subspace of _q^m^k,  _(V)=h  and ι_i(^k_i)⊈V  i∈ [t]}.
 We need the following technical lemma for the proof of the main result of this section.


Let k_1,…,k_t,k,n_1,…,n_t,m,r be positive integers, with 1≤ k_i< n_i≤ m for i ∈ [t] and k=k_1+…+k_t. Let 𝒮_i be a [n_i,k_i]_q^m/q system for i ∈ [t].
If  𝒮=𝒮_1⊕…⊕𝒮_t is (Λ_k-1,𝐤,r)-evasive, then 𝒮 is (Λ_l,𝐤,r-k+1+l)-evasive for any l ∈ [k-1].


We prove the statement by induction on k-1-l. 

If k-1-l=0, then there is nothing to prove and the assertion is clearly true. 

Suppose that l<k-1 and assume that the claim holds true for l+1.
Let us take any subspace T ∈Λ_l,𝐤.  We will show that there exists a subspace T'∈Λ_l+1,𝐤 containing T and such that

    _(T'∩𝒮)≥_(T∩𝒮)+1.

This will imply the statement, since 

    _(T∩𝒮)≤_(T'∩𝒮)-1 ≤ r-k+1+l+1-1=r-k+1+l

for every T∈Λ_l,𝐤.

We divide the proof in two cases.

Case I. First, suppose that there exists i ∈ [t] such that 

    __q^m(T∩ι_i( ^k_i))<k_i-1.

Since 𝒮_i is an [n_i,k_i]_q^m/q system contained in ι_i( ^k_i) there exists v ∈𝒮_i∖ T. Clearly, T'=T+ ⟨ v ⟩_∈Λ_l+1,𝐤 and satisfies (<ref>). 

Case II. Now assume that for any i ∈ [t] we have that 

    __q^m(T∩ι_i( ^k_i))=k_i-1.

So, l≥ k-t and T⊇ (T∩ι_1( ^k_1))+ (T∩ι_2( ^k_2))+…+(T∩ι_t( ^k_t)). 
Consider T_i=ι_1( ^k_1)+ι_i( ^k_i) for any i ∈{2,…, t}. Then, _(T_i)=k_1+k_i and 

    _(T∩ T_i)∈{ k_1+k_i-2, k_1+k_i-1}.

We further distinguish two subcases.

Case II.a.
Assume that there exists i ∈{2,…, t} such that _(𝒮∩ T_i)=k_1+k_i-2. This implies that 

    T∩ T_i=(T ∩ι_1( ^k_1))+(T ∩ι_i( ^k_i))

and there exist v_1 ∈𝒮_1 and v_i∈𝒮_i such that v_1,v_i ∉ T. Note that T∩⟨ v_1,v_i⟩__q^m={0} and v_1+v_i ∈𝒮_1⊕𝒮_i, therefore T'=T⊕ v_1+v_i__q^m∈Λ_l+1,𝐤 and satisfies (<ref>).

Case II.b. Suppose now that for every i ∈{2,…,t} we have _(𝒮∩ T_i)=k_1+k_i-1.
Since _(T∩ T_i)=k_1+k_i-1  and _(T∩ι_i( ^k_i))=k_i-1, then for each i ∈{2,…,t} there exists v_i ∈ T_i ∖ (ι_1( ^k_1)∪ι_i( ^k_i))  such that

    T=(T∩ι_1( ^k_1))+ (T∩ι_2( ^k_2))+…+(T∩ι_t( ^k_t))+ v_2,…,v_t_.

It follows that _(T)=k-1, a contradiction to the assumption that l<k-1. Thus, this subcase is not possible and this concludes the proof.




The next result is the main theorem of this section. It provides a geometric characterization of the independent spaces of the direct sum of t many uniform q-matroids.





Let k_1,…,k_t,k,n_1,…,n_t, n, m be positive integers, with 1≤ k_i< n_i≤ m for i ∈ [t] and k=k_1+…+k_t, n=n_1+…+n_t. Let _i be an -representation for _k_i,n_i(q) for each i∈ [t]. Define the  system

    :=⊕_i=1^t_i.

Moreover, for each 𝒥⊆ [t], denote by k_𝒥:=∑_j∈𝒥k_j and _𝒥:=∑_j∈𝒥ι_j(_j).
Then the following statements are equivalent.

    
  * (,ρ_) is an -representation for _k_1,n_1(q)⊕…⊕_k_t,n_t(q).
    
  * For every _q-subspace I⊆ it holds that    
    ρ_(I)=__q (I) ⟺_(I∩_𝒥)≤ k_𝒥,  ∀ 𝒥⊆[t].

    
  * 𝒮 is (Λ_k-1,𝐤,k-1)-evasive, where 𝐤=(k_1,…,k_t).



   
    (1) ⟺ (2): Let us call ℳ_1=(,ρ_) and ℳ_2=_k_1,n_1(q)⊕…⊕_k_t,n_t(q). Then, _1≅_2 if and only if there exists an invertible -linear map ψ:^n_1+…+n_t⟶ such that (_1)=(ψ(_2)). Let G be any generator matrix associated to the  system  of the form
        
    G=[ G_1   0       0;   0 G_2   ⋱;           ⋱   0;   0   ⋱   0 G_t ],

        where G_i∈^k_i× n_i, is a generator matrix associated with _i, for each i ∈ [t]. Then, using the characterization of independent spaces for _1 given in Remark <ref>, and the characterization of independent spaces for _2 derived in Theorem <ref>, together with the map 
        ψ=ψ_G as in Theorem <ref>, we conclude. 
        
  (2) ⟹ (3): Assume towards a contradiction that 𝒮 is not (Λ_k-1,𝐤,k-1)-evasive. Then, there exists an _q^m-hyperplane H̅⊆_q^m^k such that ⟨_i⟩__q^m⊈H̅ for each i∈ [t], and __q (H̅∩) ≥ k. Let T:=H̅∩. Let I⊆ T be such that __q (I) =k. Since ⟨ I⟩__q^m⊆H̅, we have that ρ_(I)=__q^m⟨ I⟩__q^m< k, hence ρ_(I)__q(I), contradicting Eq. (<ref>).

      (3) ⟹ (2): 
    First of all, we observe that the implication “⟹” in Eq. (<ref>) is always true. Indeed, if _(I∩_𝒥)>k_𝒥 for some 𝒥⊆ [t], then, since ρ__𝒥(I∩_𝒥)≤ k_𝒥, we have that I∩_𝒥 is not -independent, and hence I cannot be -independent. Therefore, ρ_(I)<_(I), yielding a contradiction.
    
    Thus, we only need to show the opposite implication. 
    Let I be an -subspace of  and let Γ:=⟨ I⟩_.   
          Observe that if Γ⊇ι_i( ^k_i) for each i ∈ [t], then Γ=^k, and ρ_(I)=k. Hence, since we have by hypothesis that _(I)=_(I∩_[t])≤ k_[t]=k, we automatically obtain equality, and ρ_(I)=k=_(I). Therefore, we may assume that Γ does not contain all the spaces ι_i( ^k_i).
        
        
        Assume by contradiction that (<ref>) is not satisfied. Then there exists an -subspace I⊆ such that _(I)≤ k, _(I∩_𝒥) ≤ k_𝒥 for each 𝒥∈ [t] and ρ_(I)<_(I). 
        We divide the proof in two cases:
        
        Case I. If Γ does not contain any of the -subspaces ι_i( ^k_i), then ℓ:=ρ_(I) <k and Γ∈Λ_ℓ,𝐤. This contradicts the hypothesis of  being (Λ_k-1,𝐤,k-1)-evasive, due to Lemma <ref>. 
        Furthermore, this also shows that for every -subspace I'⊆, if Γ':=⟨ I'⟩_ does not contain any space of the form ι_i( ^k_i) for i ∈ [t], then 
    
    ρ_(I')=min{k,_(I')}.

        Case II. 
        Up to a permutation of the set [t], we can assume without loss of generality  that Γ contains the -subspaces ι_i( ^k_i), for each i∈[a], for some  a such that 0<a<t (note that a=0 is Case I, and a=t was already proved before). Let _(I)=r+s+t with r=_(I∩_[a] )≤ k_[a] and s=_(I∩_[t]∖[a])≤ k_[t]∖[a]. I can be written as 
    I=⟨ (v_1 | 0),…, (v_r| 0)⟩_+⟨ (0  | u_1),…, (0 | u_s)⟩_+⟨ (w_1 | z_1),…, (w_t | z_t)⟩_.
 
        Moreover, the following hold:
        
            
  * X_1:=⟨ v_1,…,v_r,w_1,…,w_t⟩_⊆_[a] and _(X_1)=r+t; 
            
  * X_2:=⟨ u_1,…,u_s,z_1,…,z_t⟩_⊆_[t]∖[a] and _(X_2)=s+t. 
        
        Note that Γ=⟨ I⟩_=⊕_i ∈ [a]ι_i(_q^m^k_i)+(⟨0⟩⊕⟨ X_2⟩_),
     where ⟨ X_2⟩_ does not contain any of the spaces of the form ι_i(^k_i) for i ∈ [t]∖ [a]. Thus, by (<ref>) it must hold that 
    ρ__[t]∖[a](X_2)=min{ k_[t]∖[a],__q(X_2) }.

        Moreover, by the form of Γ, we have        
    ρ_(I)=k_[a]+_(⟨ X_2⟩_)=k_[a]+ρ__[t]∖[a](X_2).

        Combining this with the assumption that r+s+t=_(I)>ρ_(I), we obtain
        
    ρ__[t]∖[g](X_2)<s+t+r-k_[a]≤ k-k_[a]= k_[t]∖[a].

        In addition, ρ__[t]∖[a](X_2)<s+t+r-k_[a]≤  s+t =__q(X_2).
        These two last observations contradict (<ref>), since they imply ρ_𝒮_[t]∖[a](X_2)< min{ k_[t]∖[a],__q(X_2) }.






§ A REPRESENTATION OF THE DIRECT SUM OF UNIFORM Q-MATROIDS


In this section we will show that the direct sum of t many uniform q-matroids is always representable, independent of their ranks and their heights. We will do this by using  the following arguments.
First of all, we notice that, by Theorem <ref>, if such a  direct sum is representable, then it must be represented by the direct sum of the representations of its summands. Then, we  construct q-systems _1,…,_t whose direct sum satisfies Theorem <ref>(3).

We remark that the direct sums of q-systems and their connection with rank-metric codes have been studied in <cit.>, with the name of decomposable q-systems, and in <cit.>, with the name of q-systems with complementary subspaces.



 
Let k_1, … k_t, n_1,…  n_t be positive integers such that (n_i,n_j)=1 for each i≠ j,  k_i< n_i for each i ∈ [t]. Let m=n_1·…· n_t and let

    _i:={ (x,x^q,…,x^q^k_i-1)  x ∈_q^n_i},      i ∈ [t]

Then, :=_1⊕…⊕_t is a (Λ_k-1,𝐤,k-1)-evasive  system, with n=n_1+…+n_t, k=k_1+…+k_t and 𝐤=(k_1,…,k_t).


 We first introduce the following notation. For a given vector v=(v_1,…,v_ℓ)∈^ℓ, define the linearized polynomial

    v(x):=∑_j=1^ℓ v_jx^q^j-1,

and the vector v^q:=(v_1^q,…,v_ℓ^q).
Since k_i< n_i, then _i is an [n_i,k_i]_q^m/q system, for each i∈ [t]. Thus,  is an  system.
Let us prove that _(H)≤ k-1 for each hyperplane H not containing any of the spaces ι_i( ^k_i), for i ∈ [t]. Observe that such hyperplane will be of the form H=(a_1,1,…,a_1,k_1,a_2,1,…,a_2,k_2,…,a_t,1,…,a_t,k_t)^⊥, for some nonzero vectors a_i=(a_i,1,…,a_i,k_i)∈^k_i, for i∈ [t]. Equivalently, we will prove that 

    |T|≤ q^k-1,

where 

    T={(x_1,…,x_t)∈_q^n_1×…×_q^n_t∑_r=1^ta_r(x_r)=0}.


We will show this using double induction. We first start our induction on t.

If t=1, then |T|≤ q^k_1-1, since _1 is a (k_1-1)-scattered [n_1,k_1]_q^m/q system.

Now assume that the claim holds for every t'<t.
We proceed by induction on k_1. 
For k_1=1, up to dividing by a_1,1, we can assume that H=(1| a_2 |…| a_t)^⊥.
Thus,

    T={(x_1,…,x_t) ∈_q^n_1×…×_q^n_t x_1+∑_r=2^ta_r(x_r)=0 }.

Since (n_i,n_t)=1 for every i ∈ [t-1], there exists h∈ℕ such that h≡ 1 n_t and h≡ 0n_i for every i ∈ [t-1]. Therefore, if (x_1,…,x_t)∈ T, then

    ∑_r=2^ta_r(x_r)=-x_1=-x_1^q^h=∑_r=2^t-1a_r^q^h(x_r)+a_t^q^h(x_t^q).

Thus, 

    ∑_r=2^td_r(x_r)=0, 
    
    x_1+∑_r=2^ta_r(x_r)=0,

where d_r=a_r-a_r^q^h for each r ∈{2,…,t-1},
d_t,i=a_t,i-a_t,i-1^q^h for each i ∈ [k_t+1] and let a_t,k_t+1=a_t,0=0. It is readily seen that d_t=(d_t,1,…,d_t,k_t+1)≠ 0.
Let 𝒥:={i ∈{2,…,t-1} :  d_i=0}. We have two cases.

Case I: 𝒥=∅. Then, all the vectors d_i are nonzero for each i ∈{2,…,t}, and 

    T⊆ T':={(-∑_r=2^ta_r(x_r),x_2,…,x_t)∈_q^n_2×…×_q^n_t∑_r=2^td_r(x_r)=0}.

By induction hypothesis, we have |T'|≤ q^k_2+…+(k_t+1)-1=q^k_1+…+k_t-1.

Case II: 𝒥≠∅. Since d_t ≠ 0, up to reordering the variables, we may assume 𝒥={2,…,s}, for some s<t. Then, (<ref>) reads as 

    ∑_r=s+1^td_r(x_r)=0, 
    
    x_1+∑_r=2^ta_r(x_r)=0,

By inductive hypothesis, the first equation in (<ref>) has at most q^k_s+1+…+(k_t+1)-1 solutions in _q^n_s+1×…×_q^n_t. For each of these solutions (x̅_s+1,…,x̅_t), the second equation becomes

    x_1+∑_r=2^sa_r(x_s)=-∑_r=s+1^ta_r(x̅_r).

Note that (<ref>) is an affine equation, so its number of solution in _q^n_1×…×_q^n_s
is either 0 or equal to the cardinality of

    T':={(x_1,…,x_s) ∈_q^n_1×…×_q^n_s x_1+∑_r=2^sa_r(x_r)=0 }.

By inductive hypothesis, 

    |T'|≤ q^1+k_2+…+k_s-1,

hence the total number of solutions of (<ref>) is at most 

    q^k_s+1+…+(k_t+1)-1q^1+k_2+…+k_s-1=q^k_1+k_2+…+k_t-1.


Let us assume that (<ref>) holds for  every k_1'<k_1 and every k_2',…,k_t'∈ℕ.
Observe that, if a_1,k_1=0, then a_1(x_1)=a_1'(x_1), where 
b_1=(a_1,1,…,a_1,k_1-1), and

    T={(x_1,…,x_t)∈_q^n_1×…×_q^n_t b_1(x_1)+∑_r=2^ta_r(x_r)=0 }.

By inductive hypothesis with k_1'=k_1-1, we have |T|≤ q^k_1'+k_2+…+k_t-1<q^k_1+k_2+…+k_t-1. Therefore, we may assume a_1,k_1≠ 0, and, up to rescaling, we may take a_1,k_1=1. Again, define b_1=(a_1,1,…,a_1,k_1-1), and observe in this case that 

    a_1(x_1)=x_1^q^k_1-1+b_1(x_1).

Since (n_i,n_j)=1 for every i≠ j, there exists h∈ℕ such that h≡ 1 n_t and h≡ 0 n_i  for each i ∈ [t-1]. Therefore, if (x_1,…,x_t)∈ T, then

    -x_1^q^k_1-1-b_1(x_1)=∑_r=2^ta_r(x_r),
    
    -x_1^q^k_1-1-b_1^q^h(x_1)=∑_r=2^t-1a_r^q^h(x_r)+a_t^q^h(x_r^q),

where the second identity follows from raising the first identity to the q^h-th power. By subtracting the two identities, we get the equivalent system

    ∑_r=1^ta_r(x_r)=0,
    ∑_r=1^td_r(x_r)=0,

where d_r=a_r-a_r^q^h for each r ∈{2,…,t-1}, d_1=b_1-b_1^q^h, and d_t is given by
d_t,i=a_t,i-a_t,i-1^q^h for each i∈ [k_t+1] with a_t,k_t+1=a_t,0=0. As in the base step of the induction, one can verify that d_t=(d_t,1,…,d_t,k_t+1)≠ 0. We now proceed as in the base case of the induction on k_1. Let 𝒥:={i ∈ [t-1] :  d_i=0}. Again, we divide the proof in cases:

Case I: 𝒥=∅. Then, all the vectors d_i are nonzero for each i ∈ [t], and 

    T⊆ T':={(x_1,x_2,…,x_t)∈_q^n_1×…×_q^n_t∑_r=1^td_r(x_r)=0}.

Note that d_1∈^k_1-1, while d_i ∈^k_i for each i ∈{2,…,t-1} and d_t ∈^k_t+1. Hence, by induction hypothesis on k_1'=k_1-1, we have |T'|≤ q^k_1'+…+(k_t+1)-1=q^k_1+…+k_t-1.

Case II: 𝒥≠∅. Up to reordering the variables, we may assume 𝒥=[s], for some s<t (since d_t ≠ 0). Then, (<ref>) reads as 

    ∑_r=s+1^td_r(x_r)=0, 
    ∑_r=1^ta_r(x_r)=0,

By inductive hypothesis, the first equation in (<ref>) has at most q^k_s+1+…+(k_t+1)-1 solutions in _q^n_s+1×…×_q^n_t. For each of these solutions (x̅_s+1,…,x̅_t), the second equation becomes

    ∑_r=1^sa_r(x_s)=-∑_r=s+1^ta_r(x̅_r).

Note that (<ref>) is an affine equation, so its number of solution in _q^n_1×…×_q^n_s
is either 0 or equal to the cardinality of

    T':={(x_1,…,x_s) ∈_q^n_1×…×_q^n_s∑_r=1^sa_r(x_r)=0 }.

By inductive hypothesis on s<t, 

    |T'|≤ q^k_2+…+k_s-1,

hence the total number of solutions of (<ref>) is at most 

    q^k_s+1+…+(k_t+1)-1q^k_1+k_2+…+k_s-1=q^k_1+k_2+…+k_t-1.






Thanks to Theorem <ref>, we can  give a positive answer to the question on the representability of the direct sum of t uniform q-matroids.



 
    Let k_1,…, k_t, n_1, … n_t be positive integers such that k_i< n_i for each i ∈ [t]. Then, the direct sum
    
    _k_1,n_1(q) ⊕…⊕_k_t,n_t(q)

    is representable. In particular, it is represantable over every field  such that m=m_1·…· m_t for any m_1,…,m_t satisfying (m_i,m_j)=1 for each i ≠ j and m_i≥ n_i for i∈ [t].



    Let m_1,…,m_t be positive integers as in the statement, that is m_i≥ n_i for each i ∈ [t] and (m_i,m_j)=1 for each i ≠ j, and let m=m_1·…· m_t. Denote by 𝐤=(k_1,…,k_t) and by k=k_1+…+k_t. Using Theorem <ref>, we can construct 
  
    _i = {(x,x^q,…,x^q^k_i-1)  x ∈_q^m_i},      i ∈ [t],
 
    such that :=_1⊕…⊕_t is an [m_1+…+m_t,k]_q^m/q system which is (Λ_k-1,𝐤,k-1)-evasive. Then, by choosing any -subspace V_i ⊆_q^m_i, with _(V_i)=n_i for i ∈ [t], and defining 
      
    _i = {(x,x^q,…,x^q^k_1-1)  x ∈ V_i},     i ∈ [t],

    we obtain that :=_1⊕…⊕_t is an  system which is a (Λ_k-1,𝐤,k-1)-evasive, with n=n_1+…+n_t. Using Theorem <ref> we conclude.


Observe that although Theorem <ref> answers affirmatively to the question about the representability of the direct sum of any t uniform q-matroids, it does not characterize the integers m for which this direct sum is -representable. In Section <ref>, we will provide a  deeper analysis of those values in the special case of the direct sum of two uniform q-matroids of rank 1.






§ THE PARTICULAR CASE OF UNIFORM Q-MATROIDS OF RANK 1


We have seen in Theorem <ref> that the direct sum of uniform q-matroids is always representable. However, such a result does not characterize over which extension field _q^m we can have a representation. In general, this is a difficult question. In this section, we will focus on the direct sum of t many uniform q-matroids of rank 1.  We will first give some necessary conditions on m and then we will give some sufficient conditions on m ensuring that the direct sum of two uniform q-matroids of rank 1 is representable.



 §.§ Necessary conditions for the representability


In this subsection, we will  study the extension fields over which the direct sum of t many uniform q-matroids of rank 1 is representable. As we have seen before, this corresponds to the study of (Λ_t-1,(1,…,1),t-1)-evasive subspaces. By Lemma <ref>, these subspaces belong to the family of (Λ_1,(1,…,1),1)-evasive subspaces. This class of subspaces have been investigated in <cit.> and in <cit.> in terms of linear sets with complementary weights. A first result gives us some necessary conditions for the existence of (Λ_t-1,(1,…,1),t-1)-evasive subspaces; this can be derived by <cit.> and <cit.>.


    For i∈ [t], let n_i≥ 2, and let _i be an [n_i,1]_q^m/q system. Assume that 
    _1⊕…⊕_t is (Λ_1,(1,…,1),1)-evasive. Then, n_i≤m/2 for each i ∈ [t].


Therefore, we obtain the following necessary conditions for the representability of the direct sum of uniform matroids of rank 1.


    Let n_1,…,n_t≥ 2 be positive integers. If the direct sum
    
    _1,n_1(q)⊕…⊕_1,n_t(q)

    is -representable, then m≥ 2max{n_1,…,n_t}.


    Let 𝒮_i be an -representation for 𝒰_1,n_i(q) for any i ∈ [t].
    By Theorem <ref>,
    𝒮=⊕_i=1^t 𝒮_i is an -representation of 
    _1,n_1(q)⊕…⊕_1,n_t(q)

    if and only if it is (Λ_t-1,(1,…,1),t-1)-evasive. By Lemma <ref>, 𝒮 is also (Λ_1,(1,…,1),1)-evasive. So, the assertion follows by Theorem <ref>.



    The rank-metric codes associated with the direct sum of uniform q-matroids of rank 1, when representable, are the direct sum of 1-dimensional MRD codes. This latter class of codes have been studied in <cit.>, where such codes have been called completely decomposable rank-metric codes. 




 §.§ Construction for two summands over small extension fields


We now specialize to the study of the direct sum of two uniform q-matroids of rank 1. Combining some new and old results on the existence of (Λ_1,(1,1),1)-evasive subspaces we are able to prove the following result.


    Let n_1,n_2≥ 2 be two positive integers. The direct sum
    
    _1,n_1(q)⊕_1,n_2(q)

    is -representable if at least one of the following holds.
    
       
  * m is even and m≥ 2max{n_1,n_2};
       
  * m≥ n_1n_2;
       
  * m=t_1t_2 with t_1≥ n_1 and n_2≤t_1(t_2-1)/2+1;
       
  * m=t_1t_2 with t_1≥ n_1,n_2 and t_2≥ 2;
       
  * q=p^h, m=p^r, n_1+n_2-1≤m/2.
    



The reader can verify that Theorem <ref>(4) is covered by the combination of Theorem <ref>(1) and Theorem <ref>(3). However, since the constructions are different and the latter ones derive from existing results in the literature, we decided to include it anyway in the statement.


The rest of this section is dedicated to show Theorem <ref>. 



Theorem <ref> shows that establishing the _q^m-representability of the direct sum _1,n_1(q)⊕_1,n_2(q) is equivalent to find an [n_1,1]_q^m/q system 𝒮_1 and an [n_2,1]_q^m/q system 𝒮_2 such that 𝒮_1⊕𝒮_2 is a (Λ_1,(1,1),1)-evasive [n_1+n_2,2]_q^m/q system.
So, in the following we will exhibit  examples of such subspaces.
To this aim, we recall the following characterization of (Λ_1,(1,1),1)-evasiveness.


Let A,B⊆ be two -subsapces with A=n_1, B=n_2. Then, A⊕ B is (Λ_1,(1,1),1)-evasive
if and only if for each choice of a_1,a_2∈ A∖{0}, b_1,b_2∈ B∖{0} satisfying a_1b_2=a_2b_1,  it must hold that a_1=λ a_2, b_1=λ b_2 for some λ∈. 
Equivalently, A⊕ B is (Λ_1,(1,1),1)-evasive
if and only if for every a_1,a_2∈ A∖{0}, b_1,b_2∈ B∖{0} for which there exists λ∈ such that (a_1,a_2)=λ(b_1,b_2), we have that  λ∈_q.


Using Proposition  <ref> we can show a first example of (Λ_1,(1,1),1)-evasive subspace.


    Let n_1,n_2≥ 2 be two positive integers. Assume that m≥ n_1n_2, and let γ∈ be such that =(γ). 
Define

    _1   ={f(γ)  f ∈[x]_<n_1}, 
    _2    ={g(γ^n_1)  g ∈[x]_<n_2}.

Then, _1⊕_2 is a (Λ_1,(1,1),1)-evasive [n_1+n_2,2]_q^m/q system.


By Proposition <ref>, we need to show that if (f_1(γ),g_1(γ^n_1))=λ(f_2(γ),g_2(γ^n_1)), for some nonzero f_1,f_2∈[x]_<n_1,g_1,g_2∈[x]_<n_2,  then λ∈. Equivalently, by assuming g_1,g_2 being monic, our aim is to show that λ=1, that is (f_1(γ),g_1(γ^n_1))=(f_2(γ),g_2(γ^n_1)).

    Assume that (f_1(γ),g_1(γ^n_1))=λ(f_2(γ),g_2(γ^n_1)), then
    
    f_1(γ)/f_2(γ)=λ=g_1(γ^n_1)/g_2(γ^n_1),

    which in turn implies

    f_1(γ)g_2(γ^n_1)-f_2(γ)g_1(γ^n_1)=0.

For i ∈{1,2}, let us write

    f_i(x)=∑_j=0^n_1-1f_i,jx^j,      g_i(x)=∑_j=0^n_2-1g_i,jx^j.

Looking at h(γ):=f_1(γ)g_2(γ^n_1)-f_2(γ)g_1(γ^n_1) as a polynomial in γ and fixing any i_1,i_2 with 0≤ i_2<n_1, 
the coefficient of γ^i_1n_1+i_2 in h(γ) is f_1,i_2g_2,i_1-f_2,i_2g_1,i_1.

In particular, since h≤max{n_1 g_1 + f_2,n_1 g_2+ f_1}≤ n_1(n_2-1)+n_1-1<n_1n_2≤ m, h(γ)=0 implies that all the coefficients of the polynomial h(x) are 0. Thus, for each i_1∈{0,…,n_1}, i_2∈{0,…,n_2} we must have f_1,i_2g_2,i_1-f_2,i_2g_1,i_1=0. Together with the assumption of g_1,g_2 being monic, it readily follows that this implies f_1=f_2 and g_1=g_2, from which we conclude. 



We can make use of another additional strategy to construct such (Λ_1,(1,1),1)-evasive spaces in ^2 which, under certain assumptions, allows us to reduce the value of m. This is inspired by a method given in <cit.>, weakening its hypotheses.
Denote by U· V=⟨ ab  a∈ U,b ∈ V⟩_, for any two -subspaces U and V of .


    Let U,V⊆ be two -subspaces and let ξ∈^* be such that
    
        
        
  * V^q=V,
        
  * U· V∩ξ (U· V)={0}.
    
    Then, :=U⊕_2 with _2={b+ξ b^q  b ∈ V} is  a (Λ_1,(1,1),1)-evasive [n_1+n_2,2]_q^m/q system.



    Consider (a,b+ξ b^q) ∈ with a,b  0. Our aim is to show that for any λ∈, c∈ U and d ∈ V such that 
    
    (a,b+ξ b^q)=λ(c,d+ξ d^q),

    we have λ∈. 
    By the above equality we get
    
    {[              a=λ c,; b+ξ b^q=λ(d+ξ d^q). ].

    Since also c,d must be nonzero, the above system gives the following identity
    
    c(b+ξ b^q)=a(d+ξ d^q),

    that is ξ (cb^q-ad^q)=ad-cb. Since cb^q-ad^q, ad-cb ∈ U· V, we have that 
    
    cb^q-ad^q=ad-cb=0,

    from which we obtain λ∈. We  conclude by using Proposition <ref>.


As a consequence, from Lemma <ref> we recover the construction in  <cit.>.


    Let r be a proper divisor of m and let ξ∈∖_q^r. Let
    _1:=_q^r and _2:={a+ξ a^q  a ∈_q^r}. Then, _1⊕_2 is a (Λ_1,(1,1),1)-evasive [2r,2]_q^m/q system.


We can do better in some cases, when m is a prime power, using a particular family of subspaces studied in <cit.>. We do this by combining the following two technical lemmas with the previous Lemma <ref>.
Assume that we have an extension field  of , where m is a power of the characteristic p, that is, q=p^h and m=p^r. In this case, for each i ∈{0,…,p^r}, define the linearized polynomial

    p_i(x):=(x^q-x)∘…∘ (x^q-x)_i  times=∑_j=0^iij(-1)^i-jx^q^j,

and the -subspace of 

    ℱ_i:= p_i.


We can compute the dimension of the subspaces ℱ_i and establish some of their multiplicative properties.


    Let q=p^h and m=p^r, and for each i∈{0,…,p^r}, let ℱ_i:= p_i, where p_i is defined as in (<ref>). Then:
    
        
  * _(ℱ_i)=i, for each i∈{0,…,p^r}.
        
  * ℱ_i^q=ℱ_i, for each i∈{0,…,p^r}.
        
  * ℱ_i·ℱ_j⊆ℱ_i+j-1 for each i,j such that i+j≤ n+1.
    



    
        
  * This was proved in <cit.>.
        
  * Since all the coefficients of p_i are in , then one gets that p_i(α)=0 if and only if p_i(α^q)=0.
        
  * This was shown in <cit.>.
    


In the next result, we show that the existence of ξ in Lemma <ref> is always guaranteed, when the dimension of the subspace involved is at most m/2.


    Let V be an -subspace of  of dimension ℓ≤m/2. Then, there exists ξ∈^* such that V∩ξ V={0}.



    Assume by contradiction that V∩ξ V≠{0} for every ξ∈^*. Then, we must have
    
    ∑_ξ∈^*|(V∩ξ V)∖{0}|≥ (q-1)(q^m-1).

    On the other hand, an element β∈ V∖{0} belongs to ξ V∖{0} if and only if ξ =βγ^-1 for some γ∈ V∖{0}. Thus, each element in V∖{0} is counted exactly |V|-1=q^ℓ-1 times, and hence
    
    ∑_ξ∈^*|(V∩ξ V)∖{0}|=(q^ℓ-1)^2.

    This implies that
    
    (q^ℓ-1)^2≥ (q-1)(q^m-1),

    from which we can derive 2ℓ≥ m+1, contradicting the hypothesis.




Therefore, as a consequence we have the following construction of (Λ_1,(1,1),1)-evasive subspaces.


    Let p be the characteristic of _q^m, m=p^r, and let n_1,n_2 be any positive integers  such that n_1+n_2-1≤m/2. Furthermore, let ξ∈^* be such that ℱ_n_1+n_2-1∩ξℱ_n_1+n_2-1={0}.
    Define
    _1:=ℱ_n_1 and _2:={a+ξ a^q  a ∈ℱ_n_2}. Then, _1⊕_2 is a (Λ_1,(1,1),1)-evasive [n_1+n_2,2]_q^m/q system.



    Let us take U=_n_1 and V=_n_2. By Lemma <ref>(3), we have that U· V⊆_n_1+n_2-1 and (_n_1+n_2-1)=n_1+n_2-1≤m/2. Hence,  Lemma <ref> ensures the existence of ξ∈^* such that ℱ_n_1+n_2-1∩ξℱ_n_1+n_2-1={0}. In particular, due also to Lemma <ref>(2), U,V and ξ satisfy the hypotheses of Lemma <ref>, and thus we conclude.


Combining the results of this section we are able to prove Theorem <ref>.


Combining Theorem <ref>  with Proposition <ref>, <cit.>, Corollary <ref> and Corollary <ref>, we obtain the -representability of the direct sum 𝒰_1,n_1(q)⊕𝒰_2,n_2(q) under the assumptions (1)-(5) of the statement.





    Let n_2=2 and let n_1 be any positive integer greater than 1. Combining Corollary <ref> and Theorem <ref>(2), we obtain that _1,n_1(q)⊕_1,2(q) is -representable if and only if m≥ 2n_1. This generalizes the characterization result for the -representability of two uniform q-matroids of rank k_1=k_2=1 and height n_1=n_2=2 given  <cit.> (see Example <ref>).


    Let n_1,n_2≥ 3. Then, there are only finitely many degree extensions m such that we do not know yet whether the q-matroid _1,n_1(q)⊕_1,n_2(q) is -representable. Indeed, we know that _1,n_1(q)⊕_1,n_2(q) is -representable for every m≥ n_1n_2 (see Theorem <ref>(2)) and for every even m ≥ 2max{n_1,n_2} (see Theorem <ref>(1)). The only open cases are the odd m such that 2max{n_1,n_2}<m <n_1n_2, for which we can also give some answers depending on the values of n_1, n_2 and q using Theorem <ref>(3,4,5). 



    The smallest case left out by Theorem <ref> is whether the direct sum of two uniform q-matroids of rank 1 and height 3 is _q^7-representable. Using the algebra software magma <cit.>, we found examples of (Λ_1,(1,1),1)-evasive subspaces  for q ∈{2,3,4,5,7}. Theorem <ref> implies that the direct sum of two copies of 𝒰_1,3(q) is _q^7-representable for q ∈{2,3,4,5,7}.


We conclude the section by illustrating Theorem <ref> with a concrete example.


    Let n_1=7 and n_2=6. We know by Corollary <ref> that =_1,7(q)⊕_1,6(q) is not -representable for each m ∈ [14]. On the other hand,  is -representable for every m ≥ 42 (Theorem <ref>(2)) and for m ∈{16,18, 20,…,38, 40} (Theorem <ref>(1)). Moreover, using Theorem <ref>(4), we also derive that  is -representable for m ∈{21,27,33,35,39}.
    Finally, Theorem <ref>(5) implies that  is -representable also when 
    
    m=
            25     q=5^h, 
    
            29      q=29^h, 
    
            31     q=31^h, 
    
            37     q=37^h, 
    
            41     q=41^h.

    The only open cases are for m ∈{17,19,23,25^*,29^*,31^*,37^*,41^*}, where the ^* indicates that the case is open except for some values of the characteristic of the field, according to (<ref>).





§ CONCLUSIONS AND OPEN QUESTIONS


In this paper we studied the representability of the direct sum of t uniform q-matroids. By establishing a geometric characterization of the representability, we identified the critical role of evasiveness properties and cyclic flats. Our results show that the direct sum of uniform q-matroids is representable if we can find q-systems satisfying a certain evasiveness property.

We have constructed a q-system over sufficiently large fields that meets these properties, affirming that the direct sum of t uniform q-matroids is always representable. This construction not only demonstrates representability but also raises intriguing questions regarding the minimal extension field required for such representations.

Our exploration of the specific case involving the direct sum of two uniform q-matroids of rank 1 yielded more detailed insights. By leveraging the existing research on linear sets with complementary weights and on recent results on rank-metric codes, we determined more precise extension field order assumptions. 

At this point, there are some natural problems that arise.


       
  * The problem of characterizing the extensions over which the direct sum of t uniform q-matroids is representable stays widely open. We gave only some partial answers to this question in the general case. Only in the special case of t=2 uniform q-matroids of rank 1, we were able to provide more accurate results, leaving only finitely many cases unsolved. However, the problem is still open, even in the case of t uniform q-matroids of rank 1.
    
  * In order to study the problem of the representability of the direct sum of (not necessarily uniform) q-matroids  we would need a generalization of Theorem <ref>. This may allow us to use similar arguments to characterize the representability of q-matroids. 


abbrv



 Paper Title: The poset of normalized ideals of numerical semigroups with multiplicity three.




§ ABSTRACT 
    We study the poset of normalized ideals of a numerical semigroup with multiplicity three. We show that this poset is always a lattice, and that two different numerical semigroups with multiplicity three have non-isomorphic posets of normalized ideals.


[
    [
    July 31, 2024
=================





§ INTRODUCTION


Let S be a numerical semigroup, and let I,J be two ideals of S. We write I∼ J if there exists z∈ℤ (the set of integers) such that I=z+J. This binary relation is an equivalence relation. Let us denote by [I] the equivalence class of the ideal I. We can define the sum of two ideal classes as [I]+[J]=[I+J], where, as usual, I+J denotes the setwise addition of I and J (or Minkowski sum). If we denote by (S) the quotient of the set of ideals of S modulo ∼, then ((S),+) is a monoid, known as the ideal class monoid of S; its identity element is [S], the equivalence class of S. The ideal class monoid of a numerical semigroup was introduced in <cit.>, where some basic properties and its relation to antichains of gaps of the numerical semigroup were given.

It is well known that ((S),+) is isomorphic to (,+), where  denotes the set of ideals I of S such that min(I)=0 (normalized ideals). The isomorphism is precisely [I]↦ -min(I)+I. In <cit.> the concept of Kunz coordinates of normalized ideals of a numerical semigroup were introduced, and it gave rise to new bounds for the cardinality of the ideal class monoid of a numerical semigroup, as well as some closed formulas for the intersection, union, and sum of ideals. 

In <cit.>, it was shown that if S and T are semigroups such that ((S),+) is isomorphic to ((T),+), then S and T must be equal. So, one of the motivations to study the ideal class monoid, which was classifying numerical semigroups, was completed: the ideal class monoid of a numerical semigroup completely determines the numerical semigroup.


We can define on  the following relation: I≼ J if there exists K∈ with I+K=J. On (S), this relation translates to the relation [I]≼ [J] if [I]+[K]=[J] for some ideal K of S. Clearly, the posets (,≼) and ((S),≼) are order isomorphic.

In <cit.>, several properties and invariants of S were derived from the shape of the Hasse diagram of the poset (,≼), and the natural question of whether an order isomorphism between (,≼) and ([T],≼) (with T another numerical semigroup), forces S=T was proposed. In <cit.>, it was proven that if the posets (,⊆) and ([T],⊆) are order isomorphic, then S=T. As we will see in Section <ref> in very few cases the order relations ≼ and ⊆ coincide.

The aim of this paper is to study in deep detail the poset (,≼) in the case S has multiplicity three, and give an affirmative answer to the poset isomorphism problem proposed in <cit.>. To this end, we will see that S is fully determined by the number of quarks in (,≼) and their depths. We will extensively make use of the Kunz coordinates of the normalized ideals of a numerical semigroup. In doing so, we gain some knowledge on operations with ideals given by Kunz coordinates. Some auxiliary lemmas not restricted to multiplicity three are also presented.



 §.§ Supplemental online material
 Most of the results presented here took shape after observing many experiments carried out with the help of the  <cit.>  <cit.> package (see <https://github.com/numerical-semigroups/ideal-class-monoid>). The computations related to the examples in this manuscript can be found in that repository.



§ PRELIMINARIES


In this section, we recall some basic notions and results concerning numerical semigroups, ideals of numerical semigroups, posets and lattices.



 §.§ Numerical semigroups

A numerical semigroup is a co-finite submonoid of the monoid of non-negative integers under addition, denoted by ℕ in this manuscript. The co-finite condition is equivalent to saying that the greatest common divisor of the elements of the semigroup is one. 
The least positive integer belonging to S is known as the multiplicity of S, denoted by m(S), that is, m(S)=min(S^*), with S^*=S∖{0}.

Given A⊆ℕ, the smallest submonoid of ℕ that contains A is 

    ⟨ A⟩ = {∑_i=1^n a_i : n∈ℕ, a_1,…,a_n∈ A}.

Clearly, if S is a numerical semigroup, then ⟨ S⟩ =S. If A⊆ S is such that ⟨ A⟩ =S, then we say that A is a set of generators of S, or simply that A generates S. We say that A is a minimal set of generators of S if no proper subset of A generates S. It is well known that S admits a unique minimal set of generators S^*∖ (S^*+S^*) (see for instance <cit.>), moreover this set cannot have two elements congruent modulo m(S), and so the cardinality of the minimal set of generators, known as the embedding dimension of S, is always smaller than the multiplicity of S. The elements of S^*∖ (S^*+S^*) are known as minimal generators of S. It follows easily that s∈ S is a minimal generator if and only if S∖{s} is a numerical semigroup.


For a numerical semigroup S, the elements in ℕ∖ S are called gaps of S. The cardinality of ℕ∖ S is the genus of S, denoted by g(S). The largest gap of a numerical semigroup S different from ℕ is known as the Frobenius number of S, that is, F(S)=max(ℤ∖ S).

Associated to a numerical semigroup S one can define the order induced by S as a≤_S b if b-a∈ S, for any a,b∈ℤ. Minimal generators of S are precisely the elements in S^* that are minimal with respect to ≤_S.

The set of maximal elements of ℤ∖ S are known as the pseudo-Frobenius numbers of S, and this set is denoted by PF(S). Thus, f∈ℤ is in PF(S) if and only if f∉S and f+s∈ S for all s∈ S^*. The cardinality of PF(S) is called the (Cohen-Macaulay) type of S, and it is denoted by t(S).


A numerical semigroup S is irreducible if it cannot be expressed as the intersection of two numerical semigroups properly containing it. Every irreducible numerical semigroup is either symmetric (odd Frobenius number) or pseudo-symmetric (even Frobenius number). Recall that a numerical semigroup S is symmetric if for any x∈ℤ∖ S, F(S)-x∈ S, and S is pseudo-symmetric if F(S) is even and for any x∈ℤ∖ S, x≠F(S)/2, F(S)-x∈ S. A numerical semigroup is symmetric if and only if its type is one, and it is pseudo-symmetric if and only if PF(S)={F(S)/2,F(S)} (see <cit.> for more details). 

Let S be a numerical semigroup. A gap g of S is called a special gap if S∪{g} is a numerical semigroup. The set of special gaps is denoted by SG(S). It is well known that SG(S)={ g∈PF(S) : 2g∈ S} <cit.>.



 §.§ Ideals of numerical semigroups


A (relative) ideal of S is a set I of integers such that I+S=I and z+I⊆ S for some integer z. This last condition is equivalent to the existence of min(I). An ideal I is said to be normalized if min(I)=0.

The union of two ideals of a numerical semigroup is again an ideal, and the same holds for the intersection. Addition of ideals can be defined as follows. If I and J are ideals of S, then I+J={i+j : i∈ I, j∈ J} is also an ideal of S (see <cit.> for the basic properties of ideals of numerical semigroups). The set of ideals of S under this operation is a monoid, and its identity element is S.

If I is an ideal of a numerical semigroup S, then I+S=I, and so I can be expressed as I=X+S={x+s : x ∈ X, s∈ S} for some subset X of I. Such a set is known as a generating set of I. Notice that I=X+S, with X=Minimals_≤_S(I), and that every generating set of I must contain X. The set Minimals_≤_S(I) is known as the minimal generating set of S, and its elements are called minimal generators of I. Observe that there cannot be two different minimal generators congruent modulo the multiplicity of S. In particular, a minimal generating set of an ideal I of S has at most cardinality m(S). 

If X is a finite set of integers, then X+S is an ideal of S. If X={x} for some x∈ℤ, then we write x+S instead of {x}+S.

Let I∈. Then, x∈ I∖{0} is a minimal generator of I if and only if I∖{x}∈ (see <cit.>).




 §.§ Posets and lattices


A poset (P,≤) is a set equipped with a (partial) order relation (≤). Given two elements a,b in a poset (P,≤), b is a cover of a if a < b (that is a≤ b and a≠ b) and for every c such that a≤ c≤ b, then either c = a or c = b. 


A poset (P,≤) is a lattice if, for every pair of elements x,y∈ P there exist the infimum and the supremum of {x,y}; in such a case they are (binary) operations on the set P usually denoted by x∧ y and x∨ y, and referred to as meet and join, respectively. A lattice can be equivalently defined as a set equipped with two idempotent, associative, commutative and absorptive operations ∧ and ∨. In a lattice L, meets and joins of an arbitrary subset X⊆ L do not necessarily exist; if they do for every subset X⊆ L, then (L,∧,∨) is a complete lattice. The supremum and infimum of an arbitrary set X (if they exist) are sometimes indicated with ⋀ X and ⋁ X, respectively.
Obviously, every finite lattice is a complete lattice. 



A meet semilattice (S,∧) is a set S with an associative, commutative and idempotent operation ∧. Every meet semilattice induces a partial order relation defined as: x≤ y if and only if x∧ y = x. If the order ≤ has a maximum element, denoted 1, then (S,∧) is called a meet semilattice with one (1 is, equivalently, the neutral element for the operation ∧). Similarly, (S,∨) is a join semilattice if ∨ is an associative, commutative and idempotent operation on S. In this case, (S,∨) induces a partial order relation defined as x≤ y if  x∨ y = y, for all x,y∈ S. For the case the induced order ≤ has a minimum element, which we denote by 0, then (S,∨) is a join semilattice with zero (0 is the neutral element for ∨). In a join (meet, respectively) semilattice the element x∨ y (x∧ y, respectively) is the supremum (infimum, respectively) of the set {x,y} with respect to the induced order ≤.  
Given a poset (P,≤) and X⊆ P, we indicate by ↑X and ↓X the set of upper and lower bounds, respectively, of X, namely ↑X ={a∈ P  :  x≤ a,  for every  x∈ X} and ↓X ={a∈ P  :  a≤ x,  for every  x∈ X}. If X = {x}, we will write ↑x and ↓x instead of ↑{x} and ↓{x}. 

In a poset (P,≤) we will say that an element x∈ P has a unique cover if the poset ((↑x) ∖{x}, ≤) has a minimum; we will denote this minimum as x^c.

Every finite meet or join semilattice with one or zero, respectively, is indeed a lattice, as recalled in the following well known result in order theory. 

<cit.>
Let (S,∨) be a finite join semilattice with zero. Then, S is a lattice with the meet operation defined by 

    x∧ y = ⋁ (↓ x ∩↓ y).



Recall that, in a poset (P,≤) two elements x,y∈ P are incomparable (with respect to ≤) if x≰ y and y≰ x.


Let (P,≤) be a poset and x∈ P an element having a unique cover x^c. Then: 


  * for any y that is incomparable with x, it holds that ↑{x,y} = ↑{x^c,y}; 

  * for any y≰ x and y≤ x^c, x∨ y exists, in particular x∨ y = x^c.



(1) Let x^c be the unique cover of the element x∈ P and let y be incomparable with x. Suppose a∈↑{x,y}, that is, x≤ a and y≤ a. Observe that, since  x and y are incomparable, we deduce that a≠ x. Hence, a∈ (↑ x)∖{x}, and so  x^c≤ a. 
Thus, a∈↑{x^c,y}, showing that ↑{x,y}⊆↑{x^c,y}. The other inclusion is obvious as x < x^c.

(2) Let y ∈ P such that y≰ x and y≤ x^c. Then, x^c in an upper bound of {x, y}. Let z be an upper bound of {x, y}, that is, x≤ z and y≤ z. The assumption y≰ x forces x≠ z. Thus, z∈↑x∖{x}, and so x^c≤ z. This shows that x∨ y = x^c.








§ NORMALIZED IDEALS OF NUMERICAL SEMIGROUPS


Recall that a relative ideal I of a numerical semigroup S is a normalized ideal I if min(I) = 0. The set ℐ_0(S) of normalized ideals of S is always finite and forms a (complete) lattice under the operations of ∩ and ∪ (see <cit.>). Moreover, as mentioned in the introduction, it can be turned into a poset upon considering the partial order relation: I≼ J if there exists L∈ℐ_0(S) such that I + L = J. Observe that I≼ J implies I⊆ J. An ideal I is a quark if it is minimal with respect to ≼ in ∖{S}; in other words, a quark is a cover of S in the poset (,≼).

Let m be the multiplicity of S. If I is an ideal of S, and x∈ I, then x+ks∈ I for every non-negative integer k and s∈ S. It follows that the set (I)={ x∈ I : x-m∉I} generates I as an ideal, and it has precisely m elements, one per each congruence class modulo m. The set (I) is known as the Apéry set of I (with respect to m). Thus, (I)={w_0,w_1,…,w_m-1}, where w_i=min(I∩ (i+mℕ)).

Given a,b,n∈ℤ, with n≠ 0, we denote by a n the remainder of the (Euclidean) division of a by n (a n∈{0,…,n_1}), and we write a≡ b  n if n divides b-a.

As I⊆ℕ, we deduce that (I)⊆ℕ, and consequently for every i∈{0,…,m-1}, w_i=m x_i +i for some non-negative integer x_i. The tuple (x_1,…,x_m-1) is known as the Kunz coordinates of I (see <cit.>). Notice that we are omitting x_0, which is equal to 0, as 0=min(I). 

In the sequel, we will write I=(x_1,…,x_m-1)_𝒦 to denote that (x_1,…,x_m-1) are the Kunz coordinates of I.

Let n∈ℕ, and let i= n m.

Then, n=km+i for some k∈ℕ. We know that w_i is the minimum element in I congruent with i modulo m. Hence, n∈ I if and only if n≥ w_i, or equivalently k≥ x_i.  Hence, for an integer n=k m+i,

    km+i∈ (x_1,…,x_m-1)_𝕂 if and only if  x_i≤ k.

In particular, n∉I if and only if k∈{0,…,x_i-1}, and this holds for every congruence class modulo m. Thus, the number of non-negative integers not belonging to I is

    |ℕ∖ (x_1,…,x_m-1)_𝒦| = x_1+…+x_m-1.




With this in mind it is easy to show that if J is an ideal with Kunz coordinates (y_1,…,y_m-1), then

    (x_1,…,x_m-1)_𝒦⊆ (y_1,…,y_m-1)_𝒦 if and only if  (y_1,…,y_m-1)≤ (x_1,…,x_m-1)

with respect to the usual partial order on ℕ^m-1, and

    (x_1,…,x_m-1)_𝒦∩ (y_1,…,y_m-1)_𝒦    =(max({x_1,y_1}),…,max({x_m-1,y_m-1}))_𝒦,
    
        (x_1,…,x_m-1)_𝒦∪ (y_1,…,y_m-1)_𝒦    =(min({x_1,y_1}),…,min({x_m-1,y_m-1}))_𝒦.

Addition requires more effort, but can be derived by translating <cit.> to Kunz coordinates. If I+J=(z_1,…,z_m-1)_𝒦, then for every i∈{1,…,m-1}

    z_i=min({x_i_1+y_i_2+⌊i_1+i_2m ⌋ : i_1,i_2∈{0,…,m-1}, i_1+i_2≡ i m}).

where ⌊ q⌋=max{ z∈ℤ : z≤ q} for every q∈ℚ.

The idempotent elements in (,+) are precisely the oversemigroups of S <cit.>, that is, numerical semigroups T such that S⊆ T. They will play a central role in Section <ref>. We denote by  the set of oversemigroups of S.



  One may wonder when  coincides with . This question was solved already in <cit.>: = if and only if the multiplicity of S is less than or equal to two. In this case, (,≼) is a chain of length g(S)+1.


The following two technical lemmas will be handy later.


  Let S be a numerical semigroup, and let I∈, J∈. Then, I⊆ J if and only if I+J=J.
  
  
    If I+J=J, then I=I+0⊆ I+J=J. If  I⊆ J, then J=0+J⊆ I+J⊆ J+J=J, and so J=I+J.
  
  
  
    Let S be a numerical semigroup, and let I∈. For every J∈, I≼ J if and only if J=I+J. In particular, I∨ J=I+J for every J∈.
  
  
    Suppose that I≼ J, and so I+K=J for some K∈. Then, J=I+K=I+I+K=I+J. The converse is trivial. 
    
    Now, suppose that K∈ is such that I≼ K and J≼ K. Then, there exists L for which J+L=K. We now use that I+K=K, and obtain that I+J+L=I+K=K, which means that I+J≼ K. Also, I≼ I+J and J≼ I+J, which proves that I∨ J=I+J.
  

As a consequence of these two lemmas, we get the following result.

Let S be a numerical semigroup, and let I,J∈. The following are equivalent:

    
  * I⊆ J,
    
  * I+J=J,
    
  * I≼ J.



The following result will be used later, and describes the set of ideals of an oversemigroup T of S as the set of ideals in S greater than or equal to T with respect to ≼.


    Let S be a numerical semigroup and let T∈. Then, [T]=↑ T (in ).


    If I∈[T], then I+T=I and consequently T≼ I. 

    Let I∈↑T. Then, T≼ I and so there exists J∈ such that T+J=I. Thus, I+T=J+T+T=J+T=I, and so I∈[T].





 §.§ Multiplicity three


Let S be a numerical semigroup with multiplicity three, with Kunz coordinates S=(u,v)_𝒦. Then, by <cit.>, (x,y) are the Kunz coordinates of a (normalized) ideal I of S if and only if

    x≤ u,
    
      y≤ v,
    
      x+u≥ y,
    
      y+v+1≥ x.


Basically, the first two inequalities mean that S⊆ I, while the last two inequalities translate to I+S=I. 


For I=(a,b)_𝒦 and J=(c,d)_𝒦, (<ref>) particularizes to

    I+J=(min({a,c,b+d+1}), min({b,d,a+c}))_𝒦.




  As a particular instance of (<ref>), for I=(a,b)_𝒦, we obtain 
  
    I+I=(min({a,2b+1}), min({b,2a}))_𝒦.
 
  Therefore,  I+I≠ I if and only if either 2b+1<a or 2a<b (not both, since this would lead to 4b+2<2a<2b, a contradiction).


Let S be a numerical semigroup with multiplicity three and Kunz coordinates (u,v). Let I=(x,y)_𝒦 be an ideal of S. If I is an idempotent ideal, then by Remark <ref>, x≤ 2y +1 and y≤ 2x. It is easily checked that for idempotents, (<ref>) turns into: 

    x≤ u,
    
      y≤ v,
    
      x≤ 2y +1,
    
      y≤ 2x.


Figure <ref> shows the Hasse diagram of (ℑ_0(⟨ 3,13,17⟩),≼). Ideals are represented and placed according to their Kunz coordinates. Idempotents are displayed in grey.





  Let S be a numerical semigroup with multiplicity three. For all I,J∈, I+J=I∪ J.


  Let a_1,b_1,a_2,b_2∈ℕ such that I=(a_1,b_1)_𝒦 and J=(a_2,b_2)_𝒦. If I⊆ J, then by Lemma <ref>, we have that J=I+J, which leads to I∪ J=J=I+J. The same argument works for J⊆ I. Thus, we may suppose that I and J are incomparable with respect to set inclusion, which by (<ref>) means that (a_1,b_1) and (a_2,b_2) are incomparable in (ℕ^2,≤). 

  Suppose that a_1≤ a_2 and b_1≥ b_2. We know by (<ref>) that 
  
    I+J=(min({a_1,b_1,a_2+b_2+1}), min({a_2,b_2,a_1+b_1}))_𝒦.

  Also, a_1≤ a_2≤ a_2+b_2+1, 
  and so min({a_1,b_1,a_2+b_2+1})=min({a_1,b_1}). Similarly, a_1+b_1≥ a_1+b_2≥ b_2, and consequently min({a_2,b_2,a_1+b_1})= min({a_2,b_2}). Thus,
  
    I+J=(min({a_1,b_1}), min({a_2,b_2}))_𝒦=I∪ J.

  The case a_1≥ a_2 and b_1≤ b_2 follows analogously.


With this, we easily obtain the following consequence.


  Let S be a numerical semigroup with multiplicity three. Then, (,≼) is a distributive lattice with I∨ J=I∪ J, I∧ J=I∩ J, max_≼=ℕ, and min_≼=S.


  Let K∈ such that I≼ K and J≼ K. Then, I⊆ K and J⊆ K, which yields I∪ J⊆ K. By Lemma <ref>, (I∪ J)+K=K and so I∪ J≼ K. This proves that I∪ J=I∨ J.

  Analogously, one shows that I∩ J=I∧ J. The other assertions are trivial.






§ WHEN INCLUSION COINCIDES WITH THE ORDER INDUCED BY ADDITION


For a numerical semigroup S and  I,J∈, recall that I≼ J implies I⊆ J. Thus, it is natural to ask under which circumstances the order relations ≼ and ⊆ coincide. 
  

  The numerical semigroup with the least possible genus such that ⊆ and ≼ are not the same is S=⟨ 4,5,6,7⟩: {0,2}+S⊆{0,1,2}+S, while {0,2}+S⋠{0,1,2}+S (dashed line in Figure <ref>).





  Let S be a numerical semigroup. Then, ⊆ equals ≼ in  if and only if 
  
    S∈{⟨ 3,4⟩, ⟨ 3,4,5⟩, ⟨ 3,5⟩, ⟨ 3,5,7⟩}∪{⟨ 2,2k+1⟩ : k ∈ℕ}.



  Suppose that m(S)≥ 4. Then, {0,2}+S⊆{0,1,2}+S. If there exists I∈ with ({0,2}+S)+I={0,1,2}+S, then 1∈ I. But then, 1+2=3∈ ({0,2}+S)+I and 3∉{0,1,2}+S, a contradiction. Thus, ⊆ equals ≼ in  forces m(S)≤ 3. We already know that for m(S)≤ 2, we have that =, and thus by Corollary <ref>, ⊆ equals ≼.
  
  It remains to see what happens in the case m(S)=3. Suppose that 4∉S. Notice that {0,4}+S⊆{0,1}+S, since 4=1+3∈{0,1}+S. If there exists I with ({0,4} + S)+I={0,1}+S, then 1∈ I. 
  This implies that 5 ∈{0,1} + S (as 5 = 4 + 1∈({0,4} + S) + I).
  Hence, if S has multiplicity three and ⊆ equals ≼, then 5∈ S, and so ⟨ 3,5⟩⊆ S. There are only two oversemigroups of ⟨ 3,5⟩ with multiplicity three and not containing four, namely ⟨ 3, 5, 7 ⟩ and ⟨ 3, 5 ⟩ (this can be easily checked with the help of <cit.>): for all of them ⊆ and ≼ coincide in . The only case left is when ⟨ 3,4⟩⊆ S. This case yields two oversemigroups with multiplicity three,  ⟨ 3,4⟩ and ⟨ 3,4,5 ⟩, both fulfilling that ⊆ and ≼ coincide in .  


It follows from the above proposition that, for S∈{⟨ 3,4⟩, ⟨ 3,4,5⟩, ⟨ 3,5⟩, ⟨ 3,5,7⟩}∪{⟨ 2,2k+1⟩ : k ∈ℕ}, (,≼) is a lattice. However, as the next example shows, this does not cover all the cases for which (,≼) is indeed a lattice.



  One can check that for S=⟨ 3,7,8⟩, the poset (,≼) is a lattice, however we already know that in  the orders ≼ and ⊆ are not the same in virtue of Proposition <ref>. In fact, {0,4}+S⊆{0,1}+S, while {0,4}+S⋠{0,1}+S (see Figure <ref>).



Casabella proves in <cit.> that if S is a numerical semigroup such that the ideal class semigroup of S  and the ideal class semigroup of the semigroup algebra of S (K[S]) are isomorphic, then S has either multiplicity less than two or S=⟨ 3,4,5⟩ or S=⟨ 3,4⟩. By Proposition <ref>, this means that if S and K[S] have isomorphic ideal class monoids, then ≼ and ⊆ coincide in .   





§ THE POSET OF NORMALIZED IDEALS OF A NUMERICAL SEMIGROUP WITH MULTIPLICITY THREE IS A LATTICE



Next, we show that for any numerical semigroup S with multiplicity three, (,≼) is a lattice. First (and motivated by Lemma <ref>), we will focus our attention on ideals that are not idempotent.

Recall that if (a,b)_𝒦∈∖, then either 2b+1<a or 2b<b (Remark <ref>). Next, we study these two cases separately, showing that (a,b)_𝒦 has a unique cover.


Let S be a numerical semigroup with multiplicity three and let I=(a,b)_𝒦 be an ideal of S such that 2b+1<a. Then, J =(a-1,b)_𝒦 is the unique cover of I with respect to ≼.
Moreover, I+L=J+L for all L∈. 




Take the pair (a,b+t) with t=(a-1)-(2b+1). By hypothesis, 2b+1≤ (a-1), and so t∈ℕ. Let (u,v) be the Kunz coordinates of S. Notice that: 

  
  * a≤ u (first inequality if (<ref>), for (x,y)=(a,b));
  
  * b+t=(a-1)-(b+1)≤ v if and only if a-1≤ v+b+1; but a-1≤ a≤ b+v+1 (last inequality of (<ref>));
  
  * observe that b+t=(a-1)-(b+1)≤ a+u which trivially holds (as a≤ u);
  
  
  * b+t+v+1≥ a, since b+t+v+1≥ b+v+1≥ a (last inequality in (<ref>)).

Thus, (a,b+t) are the Kunz coordinates of a normalized ideal of S, say L; whence L=(a,b+t)_𝒦. Then,

    I+L=(min{a,2b+t+1},min{b,b+t,2a})_𝒦=(a-1,b)_𝒦=J.

This proves I≼ J. Notice that by (<ref>), |J ∖ I|= (a+b)-(a+b-1)=1 (as a matter of fact, J∖ I = {3a+1}). Suppose that there exists K with I⪵ K⪵ J. Then, I⊊ K⊊ J, and this would lead to |J∖ I|≥ 2, a contradiction. Hence, J covers I.

Next, we prove that if I⪵ L for some L∈, then J≼ L. Write L=(c,d)_𝒦. As I≺ L, there exists (x,y)_K fulfilling (<ref>) such that (a,b)_𝒦+(x,y)_𝒦=(c,d)_𝒦, and c=min{a,x,b+y+1} and d=min{b,y,a+x}=min{b,y} (by hypothesis b<a, and so b< x+a). We distinguish two cases depending on x<a or x≥ a.

  
  * x<a. In this setting (a-1,b)_𝒦+(x,y)_𝒦=(c,d)_𝒦, since min{a-1,x,b+y+1}=min{x,b+y+1}=min{a,x,b+y+1}.
  
  * x≥ a. If a≤ b+y+1, then 2b+1< a≤ b+y+1 and consequently b≤ y. Under these conditions, (a,b)_𝒦+(x,y)_𝒦=(a,b)_𝒦=(c,d)_𝒦, which is impossible, as we are assuming that I≠ L. Thus, a>b+y+1, and so a-1≥ b+y+1. In this case, (a-1,b)_𝒦+(x,y)_𝒦=(a,b)_𝒦+(x,y)_𝒦=(c,d)_𝒦, since min{a-1,x,b+y+1}=b+y+1=min{a,x,b+y+1}.

This concludes the proof, since in all the possible cases J≼ L. Notice that we have also shown that if I+M=L, then J+M=L, which proves the second part of the statement.



  Let S be a numerical semigroup with multiplicity three and let I=(a,b)_𝒦 be an ideal of S such that 2a<b. Then, J =(a,b-1)_𝒦 is the unique cover of I with respect to ≼.
Moreover, I+L=J+L for all L∈. 


Since 2a<b then 2a≤ b-1. Set t=b-1-2a∈ℕ. Let us consider the pair (a+t,b) and let us check that it fulfills the inequalities (<ref>), for (u,v)_S a generic element in S. Observe that
  
    
  * a+t≤ u if and only if b-1-a≤ u, which is equivalent to b≤ a+u+1. By (<ref>), third inequality, we know that b≤ a+u, and so a+t≤ u holds;
    
  * b≤ v, which holds by the second inequality of (<ref>), as I=(a,b)_𝒦 is an ideal;
    
  * a+t+u≥ a+u≥ b, in light of the third inequality of (<ref>) applied to I;
    
  * b+v+1≥ a+t if and only if b+v+1≥ b-1-a, equivalent to v+1≥ -1-a, which trivially holds.
  

  Thus, there exists L∈ such that L=(a+t,b)_𝒦. Moreover,
  
    I+L=(a+t,b)_𝒦+(a,b)_𝒦=(min{a,a+t,2b+1}, min{b,b-1})_𝒦=(a,b-1)_𝒦=J.

  In particular, I≼ J, J∖ I={3b+2} and so J covers I (arguing as in the proof of Lemma <ref>).
  
  Now, we show that for every L=(c,d)_𝒦∈ with I⪵ L, we have that J≼ L. As in the proof of Lemma <ref>, there exists (x,y) fulfilling (<ref>) such that (a,b)_𝒦+(x,y)_𝒦=(c,d)_𝒦. In particular, c=min{a,x,b+y+1}=min{a,x} (by hypothesis a<b, and so a<b+y+1) and d=min{b,y,a+x}. We distinguish between y<b and y≥ b.
  
    
  * y<b. In this setting, min{b-1,y,a+x}=min{y,a+x}=min{b,y,a+x}, and so  (a,b-1)_𝒦+(x,y)_𝒦=(c,d)_𝒦.
    
  * y≥ b. If b≤ a+x, then 2a<b≤ a+x, and a<x. In this case, (a,b)_𝒦+(x,y)_𝒦=(a,b)_𝒦=(c,d)_𝒦, contradicting that I≠ L. Hence, b>a+x, which leads to b-1≥ a+x. It follows that (a,b-1)_𝒦+(x,y)_𝒦=(a,b)_𝒦+(x,y)_𝒦=(c,d)_𝒦, because under the standing hypothesis, min{b-1,y,a+x}=a+x=min{b,y,a+x}.
  
  In any case, J≼ L and this concludes the proof.
 


By combining the two previous lemmas, we obtain the following.


  Let S be a numerical semigroup with multiplicity three, and let I∈∖. Then, I has a unique cover with respect to ≼, which we denote by I^c. Moreover, I+J=I^c+J for all J∈.


Non-idempotent ideals have a unique cover; moreover, a stronger fact holds, namely we can also prove that two different non-idempotent ideals cannot share the same cover.


  Let S be a numerical semigroup with multiplicity three, and let I,J∈∖. If I^c=J^c, then I=J.


  Let (a,b) and (c,d) be the Kunz coordinates of I and J, respectively. Recall that by Remark <ref> either 2b+1<a or 2a<b (not both) and either 2d+1<c or 2c<d. 
  
    
  * If 2b+1<a, then by Lemma <ref>, I^c=(a-1,b)_𝒦. If 2d+1<c, then J^c=(c-1,d)_𝒦. Thus, if I^c=J^c, we deduce that a=c and b=d, and so I=J. Suppose that 2c<d. Then, by Lemma <ref>, J^c=(c,d-1)_𝒦; whence a-1=c and d-1=b. As 2b+1<a, we deduce 2d-2+1<c+1, or equivalently, 2d<c+2, and so 4d<2c+4<d+4, which implies d≤ 1. If d = 1, then c = 0 (as 2c<d), hence a = 1 and b = 0, which is impossible as 2b+1<a. Similarly, also d=0 is impossible as 2c < d. Therefore, this can never be the case.
    
  * If 2a<b and 2c<d, then by Lemma <ref>, I^c=(a,b-1)_𝒦=(c,d_1)_𝒦=J^c, which clearly leads to I=J. If 2d+1<c, then we can argue as in the previous case. 
  
  This covers all possible cases, and concludes the proof.





  Let S be a numerical semigroup with multiplicity three and I=∖. If J I and J⪵ I^c, then I∨ J=I+J.


Observe that the fact that I∨ J exists follows from Lemma <ref>-(2).
As J⪵ I^c, there exists J_1,…,J_t∈ such that J_1=J, J_t=I^c and J_i+1 covers J_i for all i∈{1,…,t-1}. As J≠ I^c, we have that t>1.


If J_i∉ for all i, then we have that J_t-1^c=I^c. By Lemma <ref>, J_t-1=I, but then J≼ J_t-1=I, contradicting J⋠I. 

Therefore, J_i∈ for some i∈{1,…,t}. Let i be minimum with this property. By Lemma <ref>, J_i∨ I^c=J_i+I^c=I^c. Also, as J⋠I, we deduce that J_i⋠I. By Lemma <ref>-(2), J∨ I=J_i∨ I=I^c. Notice also that J_j∈∖ for all j<i. By applying several times Proposition <ref>, we obtain J+I^c=J_1+I^c=…=J_i+I^c. Thus, J+I^c=I^c=J∨ I. By using again Proposition <ref>, but now with I, J+I^c=J+I and we conclude that I∨ J=I^c=I+J.


With all these technical lemmas we are ready to prove that (,≼) is always a lattice for S a numerical semigroup with multiplicity three. Also, in this case, we can give an explicit formula for the description of two ideals.
 

  Let S be a numerical semigroup with multiplicity three. Then, (,≼) is a lattice. Moreover, for  I,J∈,
  
    I∨ J=
          I∪ J,  if  I  and  J  are comparable,
    
          I+J, otherwise,
  
  and 
   
    I∧ J=
          I∩ J,  if  I  and  J  are comparable,
    ⋁ (↓I ∩↓J), otherwise.




  Let I,J∈. We show that there exists I∨ J. If I≼ J, then I∨ J=J=I∪ J. Analogously, if J≼ I, then I∨ J=I=I∪ J. So, we can suppose that I and J are not comparable with respect to ≼. 

  If I∈, then by Lemma <ref>, I∨ J=I+J.

  If I∈∖, then there is a unique cover of I with respect to ≼ (by Proposition <ref>). Let I^c be this unique cover. For any J∈, not comparable with I, then ↑{I, J} = ↑{I^c, J} (by Lemma <ref>). 
 Thus, I∨ J exists if and only if I^c∨ J exists, and if so, I∨ J=I^c ∨ J. Also, by Proposition <ref>, I+J=I^c+J. We repeat this process for finitely many steps, thus producing an ascending chain I≺ I^c_1≺…≺ I^c_n of covers (more precisely, I^c_i covers I^c_i-1, for every i∈{i,… n}) until either I^c_n∈ or J≼ I^c_n. 
 
 In the first case, by Lemma <ref> we have I^c_n∨ J = I^c_n + J =  I^c_n-1 + J = … = I^c_1 + J = I + J, where all the right-side equalities follow by Proposition <ref>.  By Lemma <ref>, applied several times, I∨ J=I^c_n∨ J, and we conclude I∨ J=I+J.
 
 In the latter case, J≼ I^c_n. Obviously, J≠ I^c_n (as otherwise I≺ J). Hence, by Lemma <ref>, I^c_n-1∨ J =I^c_n-1 + J = … = I + J, where the right-hand side equalities follows from Proposition <ref> (where we are assuming that I^c_i∈∖, for all i∈{1,…, n} as otherwise we would be in the previous case). By Lemma <ref>-(1), I∨ J= I^c_1∨ J=…= I^c_n-1∨ J. Putting all this together, I+J=I^c_n-1∨ J=I^c_n∨ J=I∨ J.

The description of I∧ J follows from Theorem <ref>. Observe that, in the case I and J are comparable, that is, I≼ J, which implies I⊆ J, we have that I∧ J = ⋁(↓I∩↓J) = ⋁(↓I) = I (similarly for the case J≼ I).


For multiplicity four, it is no longer true that a normalized ideal I that is not idempotent has a unique cover. Thus, we cannot take advantage of the same strategy used in the case of multiplicity three.


 Let S=⟨ 4,7,9,10⟩, and let I=(2,2,0)_𝒦. Then, I+I≠ I and the ideals covering I are (1,2,0)_𝒦, (2,1,0)_𝒦, and (0,2,0)_𝒦.





§ QUARKS OF THE POSET OF NORMALIZED IDEALS


Let S be a numerical semigroup with multiplicity three. We already know that if S is irreducible, then  has at most two quarks <cit.>. Moreover, if S is symmetric we know that the only quark of S is {0,F(S)}+S <cit.>, and if S is pseudo-symmetric, then the quarks of  are {0,F(S)}+S and {0,F(S)/2}+S <cit.>.

By <cit.>, every numerical semigroup with multiplicity three is uniquely determined by its genus and its Frobenius number: S=⟨ 3,3g(S)-F(S),F(S)+3⟩. 
By <cit.>, the genus of S plus one equals the length of the largest ascending chain in (,≼). If S is symmetric, then its Frobenius number is twice the genus minus one (see for instance <cit.>), while if S is pseudo-symmetric, the Frobenius number of S equals twice its genus minus two <cit.>. This means that if S has at most two quarks, we can fully recover S from (,≼).

Let us focus on the case S is not irreducible. Recall that S is irreducible if and only if the cardinality of SG(S) is at most one <cit.>. We also know that SG(S)⊆PF(S), and that the type of S is at most three <cit.>. Putting all this together, we deduce that in the case S is not irreducible, the set of special gaps coincides with the set of pseudo-Frobenius elements. Let SG(S)=PF(S)={f'<f} (and so f=F(S)). In particular, S'=S∪{f'} is a numerical semigroup, and so is S=S∪{f} (notice that f=F(S) is always a special gap for any numerical semigroup different from ℕ).
 
In the next two lemmas we describe the set of normalized ideals of S that are not ideals of S' and of S (compare the first with <cit.>).


   Let S be a numerical semigroup with Frobenius number f. Then, S=S∪{f} is a numerical semigroup, and for every I∈, I∈∖[S] if and only if f∉I. 


  We already know that S=S∪{f} is a numerical semigroup. Notice that S={0,f}+S.

  If f∈ I, then S=S∪{f}⊆ I, and I=I+0⊆ I+S= I+({0,f}+S)=I+{0,f}=I∪ (f+I). By definition of Frobenius number, f+I⊆ f+ℕ⊆ S∪{f}=S. Hence, I∪(f+I)⊆ I∪S=I. Thus, I⊆ I+S⊆ I, and so I+S=I, which leads to I∈[S] Clearly, if I∈[S], then f∈S⊆ I. This proves the claim. 



Let S=(k_1,k_2)_𝒦 be a numerical semigroup with multiplicity three and PF(S)={f',f} with f'<f and 2f'≠ f. Then, S'=S∪{f'} is a numerical semigroup. Moreover, the set of ideals of ∖[S'] that contain f' is 

  
  * {(x_1,x_2)_𝒦∈ : x_1+k_1=x_2, x_1≤ k_1-1} if f'≡ 1 3,
  
  * {(x_1,x_2)_𝒦∈ : x_2+k_2+1=x_1, x_2≤ k_2-1} if f'≡ 2 3.



  
  We have seen already that under the standing hypothesis, SG(S)={f',f}, and so S'=S∪{f'} is a numerical semigroup.
  
  Notice that as f'∉S, f' is not a multiple of three. Thus, f' 3 is either one or two. Suppose first that f'≡ 1  3, and let k'=(f-1)/3, that is, f'=3k'+1. It easily follows that k'=k_1-1 and that the Kunz coordinates of S' are (k_1-1,k_2).
  
  If I=(x_1,x_2)_𝒦∈∖[S'],  then by (<ref>), x_1≤ k_1, x_2≤ k_2, x_1+k_1≥ x_2, and x_2+k_2+1≥ x_1. The condition f'∈ I means that x_1≤ k_1-1 by (<ref>). Therefore, as I∉[S'], and (x_1,x_2) already fulfills the inequalities x_1≤ k_1-1 and x_2+k_2+1≥ x_1, we deduce that x_1+(k_1-1)<x_2. If follows that, x_1+k_1=x_2. 
  
  For the other inclusion, if I=(x_1,x_2)_𝒦∈ is such that x_1+k_1=x_2 and x_1≤ k_1-1 hold, then f'∈ I (because x_1≤ k_1-1=k') and I∉[S'], because x_1+(k_1-1)=x_2-1<x_2, see (<ref>).

  The other case follows analogously.



Next, we see that if S is not irreducible, then it has exactly three quarks.


    Let S be a non-irreducible numerical semigroup with multiplicity  three and Kunz coordinates (k_1,k_2). 

    
  * If k_1≤ k_2, then the quarks of  are (k_1-1,k_2)_𝒦, (k_1,k_2-1)_𝒦, and (k_2-k_1,k_2)_𝒦.
    
  * If k_1> k_2, then the quarks of  are (k_1-1,k_2)_𝒦, (k_1,k_2-1)_𝒦, and (k_1, k_1-k_2-1)_𝒦.



    We know that under the standing hypothesis SG(S)=PF(S)={f',f} with f'<f=F(S). Also, f'≠ f/2 <cit.>, because this would mean that S is pseudo-symmetric. 
    

    By Proposition 4.17 and Lemma 5.10 in <cit.>, S'=S∪{f'}={0,f'}+S and S=S∪{f}={0,f}+S are quarks in (,≼). Since SG(S)={f',f}, S' and S are numerical semigroups and so idempotent ideals of S <cit.>.  Thus, by <cit.> the only idempotent quarks of  are {0,f'}+S and {0,f}+S, and both are unitary extensions of S. 

    Observe also that from PF(S)=-m(S)+Maximals_≤_S(Ap(S,3)) <cit.> and the fact that Ap(S,3)={0,w_1,w_2} with w_1,w_2∈ S^*, we deduce that Ap(S,3)={0,f'+3,f+3} (hence, f≢f'  3). In particular, by the definition of Kunz coordinates, S has Kunz coordinates equal to (k_1,k_2) with k_1=(f'+3-1)/3 and k_2=(f+3-2)/2 in the case f'≡ 1 3 (and thus f≡ 2 3), of k_1=(f+3-1)/3+1 and k_2=(f'+3-2)/2+1 in the case f'≡ 2 3 (and thus f≡ 1 3). Also, this means that {{0,f'}+S,{0,f}+S}={(k_1-1,k_2)_𝒦, (k_1,k_2-1)_𝒦}. 

    With the above notation, we have S=S'∖{f'}=S∖{f}. Observe that [S']∪[S]⊆. If I∈[S'], then S'+I=I and so S'≼ I; similarly, if I∈[S], then S≼ I. Thus, we are interested in the set ∖ ([S']∪[S]). 

    
    In order to describe ∖ ([S']∪[S]), we distinguish two cases, depending on (f'  3, f  3). Recall that f≢f'  3, and neither f' nor f is a multiple of three. 
    
        
  * The case (f'  3, f  3)=(1,2). Set k'=(f'-1)/3 and k=(f-2)/3. As f' < f, we deduce that k'≤ k. Also, k'=k_1-1 and k=k_2-1. 
        Let (x_1,x_2) be the Kunz coordinates of an ideal I∈∖ ([S']∪[S]). As I∈, by (<ref>), x_2≤ k_2, and as I∉[S]), by Lemma <ref>, f∉I and so x_2≥ k+1=k_2 by (<ref>). Thus, x_2=k_2. Also, by (<ref>), x_1≤ k_1.
        
         If f'∉I, then by (<ref>), k'=k_1-1<x_1. Hence, k_1-1<x_1≤ k_1, which forces x_1=k_1, and consequently I=S. 
        
        If f'∈ I, we apply Lemma <ref>-(1) to obtain that x_1+k_1=x_2 and x_1≤ k_1-1. As x_2=k_2, we deduce that x_1=k_2-k_1. It remains to show that for x_1=k_2-k_1, x_1≤ k_1-1 holds. Observe that x_1≤ k_1-1 if and only if 2k_1≥ k_2+1. By (<ref>) (or (<ref>)) applied to S, we know that 2k_1≥ k_2, so we need to ensure that 2k_1≠ k_2+1. If 2k_1=k_2+1, then 2f'=2(3(k_1-1)+1)=3(2k_1)-4=3(k_2+1)-4=3k_2-1=3(k_2-1)+2=f, which contradicts the fact that S is not pseudo-symmetric.
        
        This proves that the only ideals in ∖ ([S']∪[S]) are S and Q=(k_2-k_1,k_2)_𝒦.
        
        
        
  * The case (f'  3, f  3)=(2,1).  Set k'=(f'-2)/3 and k=(f-1)/3. As f'<  f, we deduce that k'< k. Also, k'=k_2-1 and k=k_1-1. 
        Let (x_1,x_2) be the Kunz coordinates of an ideal I∈∖ ([S']∪[S]). From I∈, by (<ref>), we obtain x_1≤ k_1, and as I∉[S], by Lemma <ref>, f∉I and so x_1≥ k+1=k_1 by (<ref>). Thus, x_1=k_1. Also, by (<ref>), x_2≤ k_2.

        If f'∉I, then by (<ref>), k'=k_2-1<x_2; whence, k_2-1<x_2≤ k_2, which forces x_2=k_2, and consequently I=S. 

        If f'∈ I, then by Lemma <ref>-(2) we deduce that x_2+k_2+1=x_1 and x_2≤ k_2-1. The first equality yields x_2=k_1-k_2-1. Now, it remains to see that k_1-k_2-1≤ k_2-1, or equivalently, 2k_2≥ k_1. By (<ref>) (or (<ref>) applied to (k_1,k_2)), we know that 2k_2+1≥ k_1, so we have to show that 2k_2+1=k_1 cannot hold. If k_1=2k_2+1, then 2f'=2(3(k_2-1)+2)= 3(2k_2)-2=3(k_1-1)-2=f-3∉S, which is impossible as we are assuming that f'∈SG(S) and so 2f'∈ S.
        
        Thus, the only ideals in ∖ ([S']∪[S]) are S and Q=(k_1,k_2-k_1-1)_𝒦.         
    
    Next, we prove that Q is a quark in . We know that =[S']∪[S]∪{S,Q}. Suppose on the contrary that there exists I≠ S with I≺ Q. In particular, I≠ Q and so either I∈[S'] or I∈[S]. Recall that by Lemma <ref>, [S']=↑S' and [S]=↑S. If I∈[S], then S≼ I≼ Q, and by Lemma <ref> we deduce that Q∈[S], a contradiction. Similarly, we obtain that I∉[S'], and this contradicts the fact that  either I∈[S'] or I∈[S].    
    
    

    Finally, observe that the case k_1≤ k_2 corresponds with (f' 3,f 3)=(1,2), while k_1>k_2 is equivalent to (f' 3,f 3)=(2,1).


Let S be a numerical semigroup and let I∈. The depth of I is the largest k such that there exists a sequence I_0,…, I_k∈ with I_0=I, I_k=ℕ, I_i≼ I_i+1, and I_i≠ I_i+1 for all i∈{0,…,k-1}. 


    Let S be a numerical semigroup and let I=(x_1,x_2)_𝒦 be an ideal of S. Then, the depth of I is x_1+x_2.


  If I is not idempotent, then by Proposition <ref>, we know that I has a single cover, and it is either of the form (x_1-1,x_2)_𝒦 or of the form (x_1,x_2-1)_𝒦. Otherwise, I is idempotent, and thus it is a numerical semigroup, and I∪{F(I)}={0,F(I)}+I covers I. Every oversemigroup of S has multiplicity three except ⟨ 2,3⟩ and ℕ. Thus, in any case the Frobenius number of I is not a multiple of three. Depending on the congruence class of F(I) modulo three, I∪{F(I)} will be either (x_1-1,x_2)_𝒦 or (x_1,x_2-1)_𝒦. 

  In this way, we construct a sequence of ideals I_0,…, I_k, with k=x_1+x_2 such that I_0=I, I_k=ℕ and I_i+1 covers I_i for all i. 
  
  Observe also that if I_0',…,I_t' is another sequence of ideals such that I_0'=I, I_t'=ℕ and I_i+1' covers I_i for all i, then I_i'⊊ I_i+1', which means that |I_i+1'∖ I_i'|≥ 1. This forces t to be upper bounded by |ℕ∖ I|, which by (<ref>) equals x_1+x_2.



If we apply this last result to S, we obtain that the depth of S is k_1+k_2, recovering in this way <cit.>.


With all these ingredients at hand, we can now solve a particular instance of <cit.>.


    Let S and T be two numerical semigroups, such that m(S)=3. If (,≼) and ([T],≼) are order isomorphic, then S=T. 


    By <cit.>, S and T have the same multiplicitity. Also, by <cit.>, S is irreducible if and only if T is irreducible.

    If S is irreducible, then by the discussion at the beginning of this section, S is completely determined by the height of (,≼) (the depth of S) and by the number of quarks. Thus, in this case S must be equal to T.

    Suppose now that S is not irreducible. Let (k_1,k_2) be the Kunz coordinates of S. Let g be the genus of S. We know that S has three quarks. By (<ref>), k_1+k_2=g. By the previous lemma two of the quarks of S have the same depth, and it is equal to g-1, while the third quark, Q, has depth equal to 2k_2-k_1 or 2k_1-k_2-1. Let us call this depth d. In the first case, g+d=3k_2 and so g+d≡ 0  3, while in the second case g+d=3k_1-1, which means that g+d≡ 2 3.

   Notice that k_1 and k_2 are solutions to one of these systems of equations: 
    
        
  * k_1+k_2=g, 2k_2-k_1=d,
        
  * k_1+k_2=g, 2k_1-k_2-1=d.
    
    In the first case (g+d≡ 0  3), we obtain k_1=(2g-d)/3 and k_2=(g+d)/3; while in the second (g+d≡ 2 3), k_1=(g+d+1)/3 and k_2=(2g-d-1)/3. 
    
    This proves that k_1 and k_2 (and thus S) are uniquely determined by g and d, and both g and d can be read from the Hasse diagram of (,≼).



    Let us prove that in the non-irreducible case, the third quark Q has depth smaller than the genus minus one of S (which is the depth of the other two quarks), that is, d<g-1, with the notation of the proof of Theorem <ref>. Notice that d≤ g-1, since the depth of S is g and Q covers S. If d=g-1, then in the first case, 2k_2-k_1= k_1+k_2-1, and so 2k_1=k_2+1. Then, 2f'=2(3(k_1-1)+1)=3(2k_1)-4=3(k_2+1)-4=3k_2-1=3(k_2-1)+2=f, which contradicts the fact that S is not pseudo-symmetric (as in the proof of Proposition <ref>). 
    In the second case, 2k_1-k_2-1=k_1+k_2-1, and so k_1=2k_2 and consequently 2f'=2(3(k_2-1)+2)=3k_1-2=3(k_1-1)+1=f, which is impossible as we are assuming that S is not pseudo-symmetric.
    




    Let S=⟨ 4, 9, 14, 19⟩. Then, [S] has three quarks two of them are idempotents. Their Kunz coordinates are (1,3,4), (2,2,4), and (2,3,3). Thus, all of them have depth eight. This means that for multiplicity four the strategy employed in this section for non-irreducible numerical semigroups is no longer valid.




§ ACKNOWLEDGMENTS

The research was carried out thanks to the hospitality offered by the Institute of Mathematics (IMAG) of the University of Granada and the staff exchange program ERASMUS+ funded by the European Community.

S. Bonzio acknowledges the support by the Italian Ministry of Education, University and Research through the PRIN 2022 project DeKLA (“Developing Kleene Logics and their Applications”, project code: 2022SM4XC8) and the PRIN Pnrr project “Quantum Models for Logic, Computation and Natural Processes (Qm4Np)” (cod. P2022A52CR). He also acknowledges the Fondazione di Sardegna for the support received by the projects GOACT (grant number
F75F21001210007) and MAPS (grant number F73C23001550007), the University of Cagliari for the support by the StartUp project “GraphNet”. Finally, he gratefully acknowledges also the support of the INDAM GNSAGA (Gruppo Nazionale per le Strutture Algebriche, Geometriche e loro Applicazioni). 

P. A. García-Sánchez is partially supported by the grant number ProyExcel_00868 (Proyecto de Excelencia de la Junta de Andalucía) and by the Junta de Andalucía Grant Number FQM–343. He also acknowledges financial support from the grant PID2022-138906NB-C21 funded by MICIU/AEI/10.13039/501100011033 and by ERDF “A way of making Europe”, and from the Spanish Ministry of Science and Innovation (MICINN), through the “Severo Ochoa and María de Maeztu Programme for Centres and Unities of Excellence” (CEX2020-001105-M).


9

ns-app A. Assi, M. D'Anna, P. A. García-Sánchez, Numerical semigroups and applications, Second edition, RSME Springer series 3, Springer, Switzerland, 2020.


b-k V. Barucci, F. Khouja, On the class semigroup of a numerical semigroup, Semigroup Forum, 92 (2016), 377–392. 

laura L. Casabella, On the class semigroup of numerical semigroups and semigroup rings, Diploma thesis, Scuola Superiore di Catania (2022), unpublished.

apery-icm L. Casabella, M. D'Anna, P. A. García-Sánchez, Apéry sets and the ideal class monoid of a numerical semigroup, Mediterr. J. Math. 21 (2024), Article No. 7 (28 pages). 

numericalsgps M. Delgado, P. A. García-Sánchez, J. Morais, NumericalSgps, A package  for  numerical  semigroups,  Version 1.3.1 dev (2023), Refereed GAP package, <https://gap-packages.github.io/numericalsgps>.

gap The GAP-Group, GAP – Groups, Algorithms, and Programming, Version 4.12.2, (2022), <https://www.gap-system.org>.

isom-icm P. A. García-Sánchez, The isomorphism problem for ideal class monoids of numerical semigroups, Semigroup Forum 108 (2024), 365–376.

Nation J. N. Nation, Notes on Lattice Theory, Lecture Notes, 2017.

multiplicity-3-4 J. C. Rosales, Numerical semigroups with multiplicity three and four, Semigroup Forum 71 (2005), 323–331.

ns J. C. Rosales, P. A. García-Sánchez, Numerical semigroups, Developments in Mathematics, 20, Springer, New York, 2009.






 Paper Title: Colorful positive bases decomposition and Helly-type results for cones.



§ ABSTRACT 
We prove the following colorful Helly-type result: Fix k ∈ [d-1]. Assume 𝒜_1, …, 𝒜_d+(d-k)+1 are finite sets (colors) of nonzero vectors in ^d. If for every rainbow sub-selection R from these sets of size at most max{d+1, 2(d-k+1)}, the system ax≤ 0,  a ∈ R has at least k linearly independent solutions, then at least one of the systems ax≤ 0,  a ∈𝒜_i, i ∈ [d+(d-k)+1] has at least k linearly independent solutions.

A rainbow sub-selection from several sets refers to choosing at most one element from each set (color).

The Helly-number max{d+1, 2(d-k+1)} and the number of colors d+(d-k)+1 are optimal.

Our key observation is a certain colorful Carathéodory-type result for positive bases.








Substrate stiffness modulates bacterial adhesion and diversity of adherent phenotypes across growth stages
    René Riedel^1, Garima Rani^1, Anupam Sengupta ^1,2^∗



^1Physics of Living Matter Group, Department of Physics and Materials Science,

University of Luxembourg, 162 A, Avenue de la Faïencerie, L-1511, Luxembourg

^2Institute for Advanced Studies, University of Luxembourg,

2, Avenue de l’Université, L-4365, Esch-sur-Alzette, Luxembourg

^∗E-mail: anupam.sengupta@uni.lu

    
=========================================================================================================================================================================================================================================================================================================================================================================================





§ INTRODUCTION


The celebrated Helly's theorem <cit.> posits that within a finite family of convex sets in ^d, if the intersection of any subfamily, consisting of at most d+1 sets, shares a common point, then all the sets in the family share at least one common point. This gem of combinatorial convexity has seen many different extensions and generalizations (see <cit.> for one of the most up-to-date surveys). A now-classical result by Lovász (see <cit.>) is the following “colorful” version of Helly's theorem:

[Colorful Helly theorem]
    Let ℱ_1, …, ℱ_d+1 be finite families of convex sets in ^d. Suppose that for any choice F_1 ∈ℱ_1, …, F_d+1∈ℱ_d+1, the intersection ⋂_i=1^d+1 F_i is non-empty. Then for some i ∈1, …, d+1, the intersection of all sets in the family ℱ_i is non-empty.


Clearly, Lovász's result implies the original Helly's theorem by taking ℱ_1 = … = ℱ_d+1 = ℱ. 

Recently, Bárány <cit.> obtained the following Helly-type result, generalizing the result of Katchalski <cit.>:


Assume A is a finite set of nonzero vectors in ^d. The homogeneous system of linear inequalities ax≤ 0, a ∈ A has at least k linearly independent solutions if and only if for every B ⊂ A whose size is at most max{d+1, 2(d-k+1)}, the system bx≤ 0, b ∈ B has at least k linearly independent solutions.


Bárány also showed that the constant m(k,d)= max{d+1, 2(d-k+1)} is optimal in the following sense. There are examples of finite subsets of ^d ∖0 such that the solution set of any subsystem corresponding to a subset of at most m(k,d) - 1 points contains k linearly independent solutions, but the solution set of the original system does not.

Suppose one has a finite collection of sets (colors). A rainbow sub-selection from these sets refers to choosing at most one element from each set (color). We note that for convenience, we do not require a rainbow sub-selection to intersect each of the original sets. For the sake of convenience, we will use [n] to denote the set {1, …, n} for a natural n.

Our foremost result is a “colorful” extension of Bárány's result:

 Fix k ∈ [d-1]. Assume 𝒜_1, …, 𝒜_d+(d-k)+1 are finite sets of nonzero vectors in ^d. If for every rainbow sub-selection R from these sets of size at most max{d+1, 2(d-k+1)}, the system ax≤ 0,  a ∈ R has at least k linearly independent solutions, then at least one of the systems ax≤ 0,  a ∈𝒜_i, i ∈ [d+(d-k)+1] has at least k linearly independent solutions.


Clearly, this theorem implies Propositionprp:helly_hom_systems, and the constant m(k,d) is optimal since it is optimal for its “monochromatic” version - Propositionprp:helly_hom_systems. Surprisingly, the number of color classes d+(d-k)+1 is optimal as well. We find this especially intriguing since usually the Helly-numbers for “colorful” and “monochromatic” Helly-type results are the same. Note that for some quantitative colorful Helly-type results <cit.>, the precise Helly-number remains unknown.

Notably, when d=k, both Propositionprp:helly_hom_systems and Theoremthm:colorful_linear_system_to_big can be derived from the Colorful Helly theorem (Propositionprp:colorful_Helly) with m(d,d) = d+1. This observation is proposed as an exercise for the reader.

Our proof of Theoremthm:colorful_linear_system_to_big mostly follows Bárány's proof of Propositionprp:helly_hom_systems with a critical distinction lying in the application of what we term the “Colorful Reay's decomposition.”

Let us recall another gem of combinatorial convexity, namely the “Colorful Carathéodory theorem” obtained by Bárány in <cit.>:

[Colorful Carathéodory theorem]

Assume S_1, …, S_d+1 are subsets of ^d. If a point p belongs to the convex hull of each of the sets S_1, …, S_d+1, then there are points s_1 ∈ S_1, …, s_d+1∈ S_d+1 such that p belongs to the convex hull of s_1, …, s_d+1.


We introduce a key observation regarding positive bases: a “colorful” extension of Reay's theorem <cit.>. This can be seen as a Colorful Carathéodory theorem for positive bases.

First, we recall some notions. Let S be a subset of a linear space. Its positive hull, denoted S, is the union of all finite linear combinations of elements of S with non-negative coefficients. In other words, S = ⋃_λ≥ 0λ S, where S stands for the convex hull of S. A set B forms a positive basis of a finite-dimensional linear space L if L =  B and removing any element from B changes the positive hull, meaning L =  B, yet L differs from B ∖{b} for every b in B.



Fix k ∈ [d]. Let d+k sets S_1, …, S_d+k⊂^d be such that for every i ∈ [d+k], S_i contains a k-dimensional linear subspace. Then there is a rainbow sub-selection ℛ from the sets S_1, …, S_d+k and its partition ℛ = R_1 ∪…∪ R_m such that


  * R_i≥R_i+1≥ 2 for every i ∈ [m-1].

  * For every i ∈ [m], R_1 ∪⋯∪ R_i is a linear subspace of ^d of dimension ∑_j ∈ [i]R_j- i, and ⋃_j ∈ [i] R_j is its positive basis.

  * ℛ = R_1 ∪⋯∪ R_m contains a k-dimensional linear subspace.



The number (of colors) d+k in Theoremthm:coloful_Reay_decomposition is precise as follows from the next lemma.


For every d ≥ k ≥ 1, there are d+k-1 sets S_1, …, S_d+k-1⊂^d such that for every i ∈ [d+k-1], S_i contains a k-dimensional linear subspace, but the positive cone of any rainbow sub-selection from S_1, …, S_d+k-1 does not contain a k-dimensional subspace.


We believe that a weaker version of Theoremthm:coloful_Reay_decomposition with the same bound d+k on the number of sets simply follows from the Colorful Carathéodory theorem (Propositionprp:colorful_Caratheodory). However, we will use the fact that the size of R_i is a monotonically decreasing function of i to prove Theoremthm:colorful_linear_system_to_big.

The following linear algebra consequence of Theoremthm:coloful_Reay_decomposition is simply a dual statement to Theoremthm:colorful_linear_system_to_big, and we believe it might be of interest.

Let A be a subset of ^d. The lineality space of A is the set A∩ (-A), which is a linear subspace of ℝ^d. It is the unique maximal dimension subspace that A contains. We use A to denote the lineality space of A.

Set h(k,d)=max{d+1, 2(k+1)} and m(k,d)=max{d+1, 2(d-k+1)}. Note that h(k,d) = m(d-k,d).

 Fix k ∈ [d-1]. Assume A_1, …, A_d+k+1 are subsets of ^d, and R ≤ k for every rainbow sub-selection R from these sets of size at most h(k,d). Then for some index i ∈ [d+k+1], A_i ≤ k.


The rest of the paper is organized as follows: In the next section, we establish our notation and introduce several key technical lemmas that will be utilized throughout the paper. Next, we will derive Theoremthm:colorful_Caratheodory_for_lineality_subspace and Theoremthm:colorful_linear_system_to_big from Theoremthm:coloful_Reay_decomposition in Sectionsec:Colorful_Helly_lineality_and_dual. In Sectionsec:coloful_Reay_decomposition, we will obtain Theoremthm:coloful_Reay_decomposition. It is the longest and most technical proof in the paper; it does not require any other results from the paper except for the well-known results discussed in the next section. In Sectionsec:optimality_helly_numbers, we will prove Lemmalem:coloful_Reay_decomposition_bound and show that it implies that the numbers of colors d+(d-k)+1 and d+k+1 in Theoremthm:colorful_linear_system_to_big and Theoremthm:colorful_Caratheodory_for_lineality_subspace are optimal. In the final section, we will obtain a weak nonhomogeneous version of Theoremthm:colorful_linear_system_to_big.




§ NOTATION, GENERAL PROPERTIES OF POSITIVE BASES AND CONES



 §.§ Notation

 We use [n] to denote the set {1, …, n} for a natural n.  The inner product of two vectors x and y of 
 ^d is denoted by xy. The linear hull of a set S is denoted by S.
 
 Recall that by a rainbow sub-selection from several sets (colors), 
 we are referring to choosing at most one element from each set (color). 
By a rainbow selection from several sets, we understand a choice of exactly one element from each set (color).
 
 Recall that h(k,d)=max{d+1,2(k+1)} and m(k,d)=max{d+1,2(d-k+1)}. 


 We recall that the polar of a set S ⊂ is defined by

    S = x ∈xs≤ 1   for all   s ∈ S.


The set S^∘∘ is called the bipolar of S.
Recall  the bipolar theorem (<cit.>):
[Bipolar theorem]
    
For any set K ⊂^d, the bipolar
K^∘∘ coincides with the closure of 
K ∪{0}.   

















 §.§ Positive basis and minimal positive basis

Let S be a subset of a linear space. Its positive hull, denoted S, 
is the union of all finite linear combinations of elements of S with non-negative coefficients.  We say that a set B is a positive basis of a linear finite-dimensional space L if L =  B, but L ≠B ∖{b} for any b ∈ B. 

Let B be a positive basis of L.
We say that a linear subspace L^' of L  is a positively spanned subspace with respect to B if B ∩ L^' is a positive basis of L^'.
Additionally, if L^' = n and B ∩ L^' = n+1, then we say that  L^' is minimal and B ∩ L^' is the minimal positive basis of L^'.
Clearly, the minimal positive basis of a non-trivial space consists of at least two vectors.

We will use next lemma proven in <cit.>.

Let B be a positive basis of a finite-dimensional space L.
Then B contains a minimal positive basis of some minimal linear subspace of L.



    
    The assertion of Theoremthm:coloful_Reay_decomposition implies that R_1 is a minimal positive basis of its positive hull.




 §.§ Convex cones

A convex cone C in ^d is  a set of the form 
a + A for some A ⊂^d. In this case, we will call a an apex of C. Note that according to our definition an apex is not necessarily unique.  Here and in the rest of  the paper,  we do not assume that the origin is   the apex of a cone!  By a non-trivial cone we understand a cone that differs from 
{0}.




We say that a convex cone  with apex at a is pointed if the cone does not contain any line. Clearly, if a convex cone is pointed, then it has a unique apex. 

The following lemma elementary follows from the separation lemma.

    
    Assume C is a closed convex  cone in ^d. If C is pointed, then C^∘ is full-dimensional in ^d.   






 §.§ Lineality space

The lineality space of A  is denoted by 
A.

The next two statements are exercises in Linear Algebra.


Let A be a subset of ^d with a non-trivial lineality space A.
Then A contains a positive basis of A.






We use ⊕ to denote the direct sum. 

Let L be a subspace of ^d and L^⊥ be its orthogonal complement.
We use P to denote the orthogonal projection onto L^⊥.
Let A be a subset of ^d such that A⊃ L.
Then A = L ⊕ P(A). In the case A = L, the cone P(A) is pointed.




§ PROOFS OF THEOREMTHM:COLORFUL_CARATHEODORY_FOR_LINEALITY_SUBSPACE AND THEOREMTHM:COLORFUL_LINEAR_SYSTEM_TO_BIG



We start by showing  how to derive Theoremthm:colorful_Caratheodory_for_lineality_subspace from Theoremthm:coloful_Reay_decomposition.

We will use the following simple arithmetic lemma.  
Recall that h(k,d)=max{d+1,2(k+1)}.

Let j,k,d be integers satisfying 1 ≤ j-1 ≤ k ≤ d-1. Then
jk/j-1+j ≤ h(k,d).

The lemma is simple and was proven  in <cit.>, so we omit the proof.



Let us continue with the proof of Theoremthm:colorful_Caratheodory_for_lineality_subspace. 
We essentially copy the proof given in <cit.> for the monochromatic version of the result.

Define L_i =  A_i, i ∈ [d+k+1], 
and t = min_i ∈ [d+k+1] L_i. 
Assume that t ≥ k+1, otherwise it is nothing to prove. 
By Theoremthm:coloful_Reay_decomposition,    there is a rainbow sub-selection ℛ 
from the sets A_1, …, A_d+k+1 and its partition ℛ = R_1 ∪⋯∪ R_m 
such that 


  * R_i≥R_i+1≥ 2 for every i ∈ [m-1].

  * For every i ∈ [m], R_1 ∪⋯∪ R_i is a linear subspace of  of dimension  
∑_j ∈ [i]R_j- i, and ⋃_j ∈ [i] R_j is its positive basis.

  * R_1 ∪⋯∪ R_m contains a (k+1)-dimensional linear subspace. 

The last property can be reformulated as  R_1 ∪…∪ R_m is a positive basis of at least (k+1)-dimensional linear space R_1 ∪⋯∪ R_m.


 Note that  R_i≥ 2 implies that 
 R_1∪…∪ R_j≥ j for every j ∈ [m].
Now, we prove by induction on j that R_1∪…∪ R_j≤ k.  

The base j=1. 
Since R_1 is a subset of ^d, R_1≤ d. Since R_1 is a minimal positive basis of R_1, R_1≤ d+1   ≤ h(k,d). By the assumption of the theorem, R_1≤ k.

We move to step j-1 → j assuming that j≥ 2. Since R_j has the smallest size among the sets R_1, …, R_j, we have 
R_j≤1/j-1∑_i ∈ [j-1]R_i. By the choice of the rainbow sets and by the induction hypothesis 

    ∑_i ∈ [j-1]R_i =  R_1 ∪⋯∪ R_j-1 + (j-1) ≤ k+(j-1).

Thus,

    j ≤R_1 ∪⋯∪ R_j = ∑_i ∈ [j-1]R_i +R_j - j
    ≤  k+(j-1)+k+(j-1)/j-1-j=jk/j-1.

Consequently, j≤jk/j-1  implying that j-1≤ k.

We can apply 
 Lemmalem:h(k,d)_arithmetic now. It implies that  

    ∑_i ∈ [j]R_j = 
    R_1 ∪⋯∪ R_j + j   
    ≤ h(k,d).

 Thus, 
by the assumption of the theorem, R_1 ∪⋯∪ R_j≤ k. 

Finally, for j=m, we get  k+1 ≤R_1 ∪⋯∪ R_m≤ k. This contradiction shows that our assumptions was wrong. We conclude t ≤ k, completing the proof of  Theoremthm:colorful_Caratheodory_for_lineality_subspace.




To prove Theoremthm:colorful_linear_system_to_big, we need the following simple technical lemma.




Assume A is a  finite set of nonzero vectors in ^d.  
We  use L^⊥ to denote the orthogonal complement of the lineality space
A, and P to denote the orthogonal projection onto L^⊥.
Denote the solution set of the homogeneous system ax≤ 0, a ∈ A by S.
Then S is a full-dimensional cone within L^⊥, and the identities

    S =  A = L^⊥∩ P(A)
 
hold.


The leftmost identity S =  A immediately follows from the definition of the polar set. 

By Propositionprp:basis_and_orth_complement, 
A =  A ⊕ P(A), and P(A) is pointed within ^d, and consequently, within L^⊥. Thus, by the standard properties of the polar transform (e.g. see <cit.>),

    A =  A ⊕ P(A) = L^⊥∩ P(A).
 
Since the cone P(A) is a pointed cone within L^⊥, 
Lemmalem:dual_of_pointed_cone implies that 
the cone L^⊥∩ P(A) is full-dimensional within 
L^⊥. 


By Lemmalem:sol_set_via_polar_of_cone,  the system ax≤ 0, a ∈𝒜_i has at least k linearly independent solutions if and  only if 
the dimension of 𝒜_i^⊥ is at least k. Or equivalently, the dimension of 𝒜_i is at most d-k.  Similarly, 
for a rainbow sub-selection R  
from the sets  𝒜_1, …, 𝒜_d+ (d-k)+1,  the system ax≤ 0,  a∈ R has at least k linearly independent solutions if and only if the dimension of R is at most d-k. 
The desired implication follows from Theoremthm:colorful_Caratheodory_for_lineality_subspace.








§ PROOF OF THEOREMTHM:COLOFUL_REAY_DECOMPOSITION


We start by reducing the problem to an easier one. 


Fix k ∈ [d]. Let d+ k sets S_1, …, S_d+k⊂^d be such that
for every i ∈ [d+k], 
S_i contains a k-dimensional linear subspace. 
Then there is a rainbow sub-selection R_1 ∪…∪ R_m from the sets such that 


  * R_i≥R_i+1≥ 2 for all i ∈ [m-1].

  * For every i ∈ [m], R_1 ∪⋯∪ R_i is a linear subspace of  of dimension  
∑_j ∈ [i]R_j- i. 

  * R_1 ∪⋯∪ R_m contains a k-dimensional linear subspace. 
 

The only difference between Theoremthm:coloful_Reay_decomposition and 
Theoremthm:coloful_Reay_decomposition_weak is that in the later one we do not state that ⋃_j ∈ [i] R_j is a positive basis of 
R_1 ∪⋯∪ R_i, i ∈ [m]. However, it is easy to see that Theoremthm:coloful_Reay_decomposition_weak implies Theoremthm:coloful_Reay_decomposition.


By Theoremthm:coloful_Reay_decomposition_weak, there is  a rainbow sub-selection R^' from  S_1, …,  S_d+k such that
R^' is a linear subspace and R^'≥ k. By Propositionprp:basis_of_lineality_subspace,  R^' contains a positive
basis, say R, of R^'. Applying   Theoremthm:coloful_Reay_decomposition_weak to d +  R^' copies of 
R, 

 we get the desired set system.


From now on, we proceed with the proof of Theoremthm:coloful_Reay_decomposition_weak.

Our proof is based on an inductive construction.  First, we  show how to choose the first rainbow set R_1, then we will show how to find a next rainbow set R_m provided that the dimension of 
R_1 ∪…∪ R_m-1 is strictly less than k. Finally, we will show that the process terminate after a finite number of steps and constructed
sets satisfy the desired properties. 


 §.§ Choice of the first rainbow set


We start with the simplest case of k=1.


Assume L is an n-dimensional vector space.
Assume that for every i ∈ [n+1], a finite set B_i is a
is a positive basis of a non-trivial subspace H_i of L.
Then there is a rainbow  selection R from the sets B_1, …, B_n+1. 
such that  R is a non-trivial subspace of L.


Fix i ∈ [n+1]. Since B_i is a positive basis of H_i, 0 ∈B_i. Thus, by the colorful Carathéodory theorem  – Propositionprp:colorful_Caratheodory, 
there is a rainbow selection R such that 
0 ∈R. Since the original sets are positive bases in the corresponding subspaces, they do not contain the origin. Thus, the origin belongs to the relative interior of a face F of R  satisfying F ≥ 1. It means that
R ⊃F. The lemma follows. 


Using Propositionprp:basis_of_lineality_subspace, Propositionprp:basis_contains_minimal_subbasis, and 
Lemmalem:coloful_pos_bases_helly_k=1 together, we get

Assume L is an n-dimensional vector space.
Assume that for every i ∈ [n+1], a set S_i satisfies 
S_i≥ 1.
Then there is a non-empty rainbow  sub-selection R from the sets  S_1, …, S_n+1  such that  R is a minimal basis of the minimal  subspace R. 



By Corollarycor:existence_rainbow_basis, there are a non-empty subset I of [d+k]  and a rainbow partition R from ⋃_i ∈ I S_i such that  R is a minimal positive basis of the minimal  subspace R. 
Let I_1 ⊂ [d+k]   be an index set of maximal cardinality such that  there is   a rainbow sub-selection R_1  from the union  ⋃_i ∈ I_1 S_i such that R_1 is a minimal positive basis of the minimal subspace R_1. 

Note that automatically: 

    R_1≥ 2,  R_1 =  R_1,   and   R_1 = R_1 -1.









 §.§ Inductive construction


Assume that for some m^' > 1, the sets R_1, …, R_m^'-1 has been already chosen.



If the dimension of R_1 ∪…∪ R_m^'-1 is at least k, we terminate the construction process.

 If the dimension of R_1 ∪…∪ R_m^'-1 is strictly less than k, we proceed as follows.

Denote R_1 ∪…∪ R_m^'-1 by L_m^'-1.
We use C_m^'-1 to the denote the orthogonal complement of L_m^'-1, and P_m^'-1 to denote the orthogonal projection onto C_m^'-1. 
We set 
I_m^'⊂ [k+d] ∖I_1 ∪…∪ I_m^'-1 
to be a non-empty index set of maximal cardinality such that  there is  a rainbow sub-selection R_m^' from the sets

  S_i, i ∈ I_m^',
  such that
P_m^'-1 R_m^' is a minimal positive basis  of the minimal linear subspace P_m^'-1 R_m^'. 



 §.§ Correctness of the inductive construction

We divide the proof in several simple steps. 




Now we show that it is possible to add a new rainbow set if the dimension of linearity space of the union of already chosen ones is strictly less than k.


Assume that for some m^' > 1, the sets R_1, …, R_m^'-1 has been already chosen in such a way that


  * R_1 ∪…∪ R_m^'-1 < k.

  * R_i≥  2 for each i ∈ [m^'-1].

  * For every i ∈ [m^'-1], R_1 ∪⋯∪ R_i =  ∑_j ∈ [i]R_j- i. 

 Then there are an index set I ⊂ [k+d] ∖I_1 ∪…∪ I_m^'-1 and a rainbow sub-selection 
 R from the sets S_i, i ∈ I, such that
P_m^'-1 R is a minimal positive basis  of the minimal linear subspace P_m^'-1 R.


For any i, the dimension of the space P_m^'-1( S_i) is at least
S_i - R_1 ∪…∪ R_m^'-1≥ 1.
That  is, P_m^'-1( S_i) is a non-trivial subspace of
C_m^'-1 of dimension at least d - R_1 ∪…∪ R_m^'-1. 
On the other hand, the cardinality of the set [k+d] ∖I_1 ∪…∪ I_m^'-1 is equal to  

    d+k - ∑_j ∈ [m^' -1]R_j = d - ∑_j ∈ [m^' -1]R_j - (m^' -1) + (k - (m^' -1)),

which is equal to 
d - R_1 ∪…∪ R_m^'-1 + (k - (m^' -1)) by our assumption. Since R_i≥ 2 for all i ∈ [m^'-1], we get  

    k > R_1 ∪…∪ R_m^'-1 = 
    ∑_j ∈ [m^' -1]R_j - (m^' -1) ≥  m^' -1.

Thus, by Corollarycor:existence_rainbow_basis, there are a non-empty subset I of [k+d] ∖I_1 ∪…∪ I_m^'-1  and a rainbow sub-selection  R^' from the sets P_m^'-1 (S_i), i ∈ I,  such that 
R^' is a minimal positive basis  of the minimal linear subspace 
R^'. There are s_i ∈ S_i, i ∈ I such that 
R^' =  ⋃_i ∈ I P_m^'-1 (s_i). We complete the proof by taking R  =  ⋃_i ∈ Is_i.



Assume that for some m^' > 1, the sets R_1, …, R_m^'-1 has been already chosen by our inductive construction in such a way that
 R_1 ∪…∪ R_m^'-1 < k. Then there are
 I_m^'⊂ [k+d] ∖I_1 ∪…∪ I_m^'-1 of maximal cardinality and a rainbow sub-selection  R_m^' from S_i, i ∈ I_m^', 
such that P_m^'-1 R_m^' is a minimal positive basis  of the minimal linear subspace P_m^'-1 R_m^'. 


 

Now let us show that the other desired properties of R_m^' hold.

We will need the following lemma, which is essentially Lemma 2.5 of <cit.>.


 Let S be  a  subset of a  linear space. Assume 


  * There is  a subset  R  of S of maximal cardinality 
 such that R is a minimal positive basis of the minimal  subspace R.

  * There is  a subset  Q  of S ∖ R of maximal cardinality 
 such that  P Q is a minimal positive basis of the minimal subspace PQ, where P denotes
the orthogonal projection onto the orthogonal complement of R.
 
Then
Q≤P, 
R ∪ Q = R ∪ Q = R ∪ P(Q) ,
and   R ∪ Q = Q +  P - 2.


Note that R and Q are disjoint since P(R) = 0 and P(Q) is a positive basis of a subspace of the image of P.  Denote I = [ R] and 
J = [ Q]. Let R = {r_1, …, r_R} and 
Q = {q_1, …, q_Q}.
Since  P Q is a minimal positive basis of the minimal subspace PQ, there is a unique positive convex combination 
∑_j ∈ Jβ_j P(q_j) that equals to zero. Since P is the projection onto orthogonal complement of R, one has ∑_j ∈ Jβ_j q_j ∈ R.  

Assume ∑_j ∈ Jβ_j q_j = 0. Then the origin is in the relative interior of  Q, and therefore Q is a linear subspace of dimension Q - 1. In this case the choice of R implies the inequality 
R≥Q. Moreover, in this case, the subspace R ∪ Q is equal to the direct sum of the linear subspaces R and Q. Hence,
R ∪ Q = R +  Q - 2 and R ∪ Q is a positive basis of R ∪ Q.


Assume now that ∑_j ∈ Jβ_j q_j ≠  0. 
Since R is a minimal basis of R,  there is a unique  proper  subset I^' of I of minimal cardinality and positive coefficients α_i, i ∈ I^', such that 

    ∑_j ∈ Jβ_j q_j + ∑_i ∈ I^'α_i r_i = 0.

This equation and the minimality of I^' implies that the origin is in the relative interior of R^'∪ Q, where R^' = {r_i   i ∈ I^'}.
Hence, R^'∪ Q =  R^'∪ Q.
However, the construction shows that 
R^'∪ Q = I^'+ Q  - 1. Hence R^'∪ Q is the minimal positive basis for  
R^'∪ Q. By the choice of R, 
R≥ Q ∪ R^' > Q.

Since R =  R by the choice of R, we have

    R ∪ Q = R ∪ P(Q).

Clearly, P(Q) = Q. However, R∪ P(Q) is the direct sum of R and P(Q), which is a linear subspace of dimension R + Q -2 by the choice of R and Q. 
 The proof of 
Lemmalem:Reay_property_of_iteration_step is complete.



Assume that for some m^' > 1, the sets R_1, …, R_m^'-1 has been already chosen by our inductive construction in such a way that


  * R_1 ∪…∪ R_m^'-1 < k.

  * R_i≥  2 for each i ∈ [m^'-1].

  * For every i ∈ [m^' -1], R_1 ∪⋯∪ R_i is a linear subspace of  of dimension  
∑_j ∈ [i]R_j- i. 

 Then the inductive construction described in Subsectionsubsec:next_rainbow_set returns  R_m^' satisfying the properties:
 

  * R_m^'≥  2 for each i ∈ [m^'-1].

  * R_m^'-1≥R_m^'.

  * R_1 ∪⋯∪ R_m^' is a linear subspace of  of dimension  
∑_j ∈ [i]R_j- m^'. 




By Corollarycor:existence_of_the_next_rainbow_set, the inductive construction described in Subsectionsubsec:next_rainbow_set returns  R_m^'. By construction, R_m^'≥ 2. 

We use P_0 to denote the identity operator on ^d, L_0 and C_0 to denote 
0 and ^d, respectively.  
Set
R = P_m^' -2 (R_m^' -1)  and Q = P_m^' -2 (R_m^'). 
Since C_m^' - 2⊃ C_m^' - 1⊃ C_m^', 

    P_m^' -1(Q) = P_m^' -1P_m^' -2(R_m^') = P_m^' -1 (R_m^').
 
Hence, 
R = R_m^' -1, Q = R_m^'. 
Moreover, R is a minimal basis of  R by the choice of R_m^' -1.
Thus, the sets R and Q satisfy the assumptions of Lemmalem:Reay_property_of_iteration_step with S = R ∪ Q. 
Consequently, R_m^' -1≥R_m^' and 
R ∪ Q is a linear space of the dimension R_m^' -1 + R_m^' -2.  
Hence, by Propositionprp:basis_and_orth_complement,

    L_m^' = L_m^' -2∪ R_m^' -1∪ R_m^' = 
    L_m^' -2∪  P_m^' -1R_m^' -1∪  P_m^' -1R_m^' =


    L_m^' -2⊕ P_m^' -1R_m^' -1∪  P_m^' -1R_m^' = 
     L_m^' -2⊕R ∪ Q.

Thus, L_m^' is a linear space of the dimension 

    ∑_j ∈ [m^'-1]R_j- m^' +2 +  R_m^' -1 + R_m^' -2 = ∑_j ∈ [m^']R_j- m^'.

This completes the proof of the lemma. 



 
Let us combine together all the small lemmas to prove Theoremthm:coloful_Reay_decomposition_weak.

Choose R_1 as described in Subsectionsub:first_rainbow_set.
The existence of such a choice was justified in Subsectionsub:first_rainbow_set.

 If R_1 > k, then the inductive construction
terminates immediately at m = 1 and by (<ref>) R_1 is the desired rainbow sub-selection.  

If R_1≤ k, we use the inductive construction from Subsectionsubsec:next_rainbow_set to construct sets R_2, …, R_m while
R_1 ∪…∪ R_m-1  < k. 
By Corollarycor:existence_of_the_next_rainbow_set and (<ref>), we can choose the next rainbow set as described the inductive construction. By Lemmalem:properties_of_the_rainbow_next_set,
the constructed set system satisfies the desired properties. Since we add at least one dimension at each step, the process terminates after at most k steps.  




§ OPTIMALITY OF HELLY-NUMBERS


We start by proving Lemmalem:coloful_Reay_decomposition_bound, which shows the optimality of d+k in Theoremthm:coloful_Reay_decomposition.


We will use induction on d. 

For d=k, set S_1 = … = S_2k-1 = ± e_1, …, ± e_k. Any rainbow sub-selection R from these sets does not contain at least one of the 2k points ± e_1, …, ± e_k. Hence, R cannot contain the whole ^k.

Now, we show how to add one dimension and a new set to our set system. We assume that ^k is the subspace of ^d spanned by its first k standard basis vectors e_1, …, e_k for any k ∈ [d]. 

The inductive construction is as follows: Assume that for some d > k, we have found sets S_1, …, S_d+k-2⊂^d-1. Then we set S_d+k-1 to be any positive basis of ^d whose vectors do not belong to ^d-1. 

Let us show that the inductive construction returns the desired sets. To be more precise, we will show that R = r ∈ R ∩^k for any rainbow sub-selection R. 

Note that S_1 = … =  S_2k-1 = ^k and S_2k-1 +i = ^k+i.

The case d=k was already considered. Assume d > k, and assume that for any rainbow sub-selection R_d-1 from ⋃_i ∈ [d+k-2] S_i, the identity 

    R_d-1 = r ∈ R_d-1∩^k

holds.
 
Consider an arbitrary R = r_1, …, r_d+k-1 with r_i ∈ S_i, i ∈ [d+k-1]. Set R_d-1 = R ∖r_d+k-1. We claim R =  R_d-1. Clearly, R ⊃ R_d-1. Assume that R ⊄ R_d-1. Then there is u such that u ∈ R and u ∉ R_d-1. The latter implies that at least one of the vectors u or -u does not belong to R_d-1. This, the choice of our set system, and the inclusion u ∈ R imply that ue_d+k-1≠ 0. However, the inclusion u ∈ R ⊂ R yields -u ∈ R ⊂ R. Since R contains exactly one vector with a non-zero last coordinate, the sign of all non-zero last coordinates of the vectors of R coincide with that of r_d+k-1. That is, R cannot contain both u and -u, a contradiction. Thus, R =  R_d-1.

Since r_d+k-1∉^k by construction, we get

    R =  R_d-1 = r ∈ R_d-1∩^k = r_1, …, r_2k-1.

By the case d=k, R < k-1. Clearly, the same inequality is true for any subset of R. Thus, the proof of Lemmalem:coloful_Reay_decomposition_bound is complete.



The sets S_i, i > 2k-1, can be chosen in such a way that S_i = k.



The number of colors d + k + 1 in Theoremthm:colorful_Caratheodory_for_lineality_subspace is optimal. For any k ∈ [d-1], there are subsets A_1, …, A_d+k with A_i > k, i ∈ [d+k], and such that ^d, and R ≤ k for every rainbow sub-selection R from these sets.



We do not assume that the size of R is at most h(k,d). 



Since k ∈ [d-1], 1 < k+1 ≤ d. Thus, any d+k sets returned by Lemmalem:coloful_Reay_decomposition_bound for k → k+1 provide the desired example.


Similarly, one gets


The number of colors d + (d-k) + 1 in Theoremthm:colorful_linear_system_to_big is optimal. For any k ∈ [d-1], there are 𝒜_1, …, 𝒜_d+ (d-k) finite subsets of nonzero vectors in ^d such that the dimension of the solution set of the systems ax≤ 0,  a∈𝒜_i, i ∈ [d+ (d-k)] is strictly less than k, but for every rainbow sub-selection R from these sets the system ax≤ 0,  a∈ R has at least k linearly independent solutions.



Since k ∈ [d-1], 1 < d-k+1 ≤ d. By Lemmalem:sol_set_via_polar_of_cone, any d+(d-k) sets returned by Corollarycor:optimality_lineality_subspace for k → d-k provide the desired example.


As mentioned in the Introduction, the optimality of h(k,d) and m(k,d) was shown in <cit.>. For completeness, we provide the example that shows the optimality of h(k,d) (the optimality of m(k,d) follows immediately from it and Lemmalem:sol_set_via_polar_of_cone).

Consider the case d+1 ≥ 2(k+1) in which h(k,d) = d+1. Set A to be the vertices v_1, …, v_d+1 of a regular simplex with v_1 + … + v_d+1 = 0. Then, A = ^d and for any B ⊂ A of size at most d, B = {0}.

Consider the case d+1 < 2(k+1) in which h(k,d) = 2(k+1). Set A to be the set of vertices of the (k+1)-dimensional cross-polytope ± e_1, …, ± e_k+1. Clearly, one has A = k+1 but A ∖a≤ k for every a ∈ A.



§ NONHOMOGENEOUS LINEAR SYSTEMS


Propositionprp:helly_hom_systems was a byproduct of the proof method of the following result <cit.>:

Fix k ∈ [d-1]. Assume ℱ is a finite family of convex sets in ^d. If the intersection of any subfamily of at most m(k,d) sets contains a k-dimensional cone, then the intersection ⋂_F ∈ℱ F of all sets in the family contains a k-dimensional cone.

Let us show that Propositionprp:helly_cones and Propositionprp:helly_hom_systems are equivalent. Using a somewhat standard reduction procedure, one can equivalently assume that ℱ is a finite family of the solution sets of finite nonhomogeneous systems of linear inequalities. The following trivial statement allows us to move a convex cone inside a convex set:

Let K ⊂^d be a closed convex set containing a cone C with apex at u. Then K contains the cone v - u + C for every v ∈ K; v is an apex of this cone.


Using the fact that m(k,d) ≥ d+1 in the classical Helly's theorem and Lemmalem:apex_shift, we see that one can equivalently assume that ℱ is a finite family of the solution sets of finite homogeneous systems of linear inequalities. Thus, Propositionprp:helly_cones and Propositionprp:helly_hom_systems are indeed equivalent.

Unfortunately, the previous argument does not apply directly in the colorful case. We can show the following weaker colorful version of Propositionprp:helly_cones, which is essentially a nonhomogeneous version of Theoremthm:colorful_linear_system_to_big.


Fix k ∈ [d-1]. Assume ℱ_1, …, ℱ_d+(d-k)+1 are finite families of convex sets in ^d such that the intersection of sets in any rainbow sub-selection R of size at most m(k,d) from these sets contains a k-dimensional cone. Then for some i ∈ [d+(d-k)+1], all sets in ℱ_i contain a translate of some k-dimensional cone C.

Our proof strategy is to reduce the problem to the homogeneous case obtained in Theoremthm:colorful_linear_system_to_big.

We need the following two properties of convex cones.


Assume C is a closed convex cone in ^d with apex at the origin, and c_1, …, c_n are points of C. Then

    c_1 + … + c_n + C ⊂⋂_i ∈ [n]c_i + C.



The lemma follows from Lemmalem:apex_shift and the observation that c_1 + … + c_n - c_i ∈ C for every i ∈ [n].



Assume K is a closed convex set in ^d and a non-trivial closed convex cone C with apex at the origin is contained within K^∘∘. Then K contains a translate of C.


Since K^∘∘ contains a point different from the origin, K is non-empty; say p ∈ K. Take any vector v within C and t > 2. Since t v belongs to the closure of K ∪{0}, there is a vector v_t in K such that the segment with endpoints 0 and v_t intersects the unit ball centered at t v, that is, λ_t v_t = t v + δ_t with δ_t≤ 1 and λ_t ∈ [0,1]. By convexity, the point p + λ (v_t - p) is in K for all λ∈ [0,1]. Thus,

    K ⊃ p + λ_t/t (v_t - p) = p 1 - λ_t/t + t v + δ_t/t→ p + v   as   t →∞.

Hence, p + v ∈ K, and consequently, p + C ⊂ K.


Now, we are ready to prove Theoremthm:colorful_helly_cones, a non-homogeneous version of Theoremthm:colorful_linear_system_to_big.


The idea is to reduce the original families to families of closed polyhedral convex cones with apexes at the origin, that is, to the situation resolved in Theoremthm:colorful_linear_system_to_big.

Without loss of generality, we can assume that all the sets are closed. Indeed, a convex set contains a non-trivial cone if and only if the closure of the set contains the closure of the cone. Also, we disregard the sets that coincide with the whole space ^d.

For every rainbow sub-selection R from the families ℱ_1, …, ℱ_d+(d-k)+1, we use C(R) to denote a k-dimensional cone of the form v_1(R), …, v_k(R) with linearly independent vectors v_1(R), …, v_k(R) such that its translate is contained in the intersection set of R.

For every i ∈ [d+(d-k)+1] and every F_i ∈ℱ_i, define F_i^' as the convex hull of the cones C(R) for all the rainbow sub-selections R of size at most m(k,d) from ℱ_1, …, ℱ_d+(d-k)+1 containing F_i. Since K and the origin are contained within K^∘∘, Lemmalem:apex_shift implies that every such C(R) is contained in F_i^∘∘. Hence, F_i^'⊂ F_i^∘∘. Also, F_i^' is a polyhedral cone with an apex at the origin, that is, F_i^' is the solution set of a finite homogeneous system of linear inequalities a_ix≤ 0, a_i ∈ A_i ⊂^d ∖{0}.

Define ℱ_1^', …, ℱ_d+(d-k)+1^' by

    ℱ_i^' = F_i^' F_i ∈ℱ_i,      i ∈ [d+(d-k)+1].

On the one hand, we have that the intersection set of any rainbow sub-selection R^' of size at most m(k,d) from the families ℱ_1^', …, ℱ_d+(d-k)+1^' contains a k-dimensional cone with an apex at the origin. On the other hand, for every i ∈ [d+(d-k)+1], the intersection set of ℱ_i^' can be described as the solution set of a finite homogeneous system of linear inequalities a_ix≤ 0, a_i ∈𝒜_i ⊂^d ∖{0}. The sets 𝒜_1, …, 𝒜_m(k,d) satisfy the assumption of Theoremthm:colorful_linear_system_to_big. Thus, for some i ∈ [d+(d-k)+1], the system ax≤ 0,  a∈𝒜_i, has k linearly independent solutions, that is, the intersection ⋂_F_i^'∈ℱ_i^' F_i^' contains a k-dimensional cone with an apex at the origin. As shown, this means that the intersection ⋂_F_i ∈ℱ_i F_i^∘∘ contains a k-dimensional cone with an apex at the origin.

Applying Lemmalem:cone_containment_set_bipolar, we see that each of the sets of ℱ_i contains a translate of some k-dimensional cone. The proof of Theoremthm:colorful_helly_cones is complete.


We believe that under the assertion of Theoremthm:colorful_helly_cones, there is a family whose intersection set contains a k-dimensional cone.






alpha





 Paper Title: Saturation of edge-ordered graphs.


	
	Multi-label Sewer Pipe Defect Recognition with Mask Attention Feature Enhancement and Label Correlation Learning
    Vladimir BoškovićUniversité Paris-Saclay, CNRS, CEA, Institut de Physique Théorique, 91191 Gif-sur-Yvette, France
Email address: vladimir.boskovic@ipht.fr Balázs KeszeghHUN-REN Alfrd Rnyi Institute of Mathematics and ELTE Etvs Lornd University, Budapest, Hungary.
Email address: keszegh@renyi.hu
			
    August 5, 2024
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§ ABSTRACT  
		For an edge-ordered graph G, we say that an n-vertex edge-ordered graph H is G-saturated if it is G-free and adding any new edge with any new label to H  introduces a copy of G. The saturation function describes the minimum number of edges of a G-saturated graph. In particular, we study the order of magnitude of these functions. For (unordered) graphs, 0-1 matrices, and vertex-ordered graphs it was possible to show that the saturation functions are either O(1) or Θ(n). We show that the saturation functions of edge-ordered graphs are also either O(1) or Ω(n). However, by finding edge-ordered graphs whose saturation functions are superlinear, we show that such a dichotomy result does not hold in general. 
		
		Additionally, we consider the semisaturation problem of edge-ordered graphs, a variant of the saturation problem where we do not require that H is G-free. We show a general upper bound O(n log n) and characterize edge-ordered graphs with bounded semisaturation function. 
				
		We also present various classes of graphs with bounded, linear and superlinear (semi)saturation functions. Along the way, we define a natural variant of the above problem, where the new edge must get the smallest label. The behaviour of the two variants shows many similarities, which motivated us to investigate the second variant extensively as well.
		
		
	
	
	

§ INTRODUCTION

	
	An edge-ordered graph is a finite simple graph G with a linear order on its edge set E. We usually assign the edge-order using an injective labeling l: E →ℕ, but it can equivalently be any other linear order, e.g., l: E →ℝ. We say that an edge-ordered graph H contains an edge-ordered graph G, if G is isomorphic to a subgraph of H. Note that, an isomorphism between two edge-ordered graphs has to respect the edge-order. If H does not contain G, then we say that H avoids G. 
	
	A classical saturation problem of (not edge-ordered) graphs asks for the minimum number of edges in an n-vertex graph H such that it avoids G and adding any missing edge to H, a copy of G is created. In general, we always assume that n is a large enough integer, and we denote by sat(n, G), the saturation number, the number of edges in such a graph H. 
	
	In the extremal problem of edge-ordered graphs <cit.> we are looking for the maximal size saturating host graph in the sense that no edge can be added with any label:
	
	 
		We are given an edge-ordered graph G.		
		Let ex_e(n,G) be the maximum number of edges in an edge-ordered graph H on n vertices that does not contain G as a subgraph (respecting the edge-order).
	
	
	
	We extend the definition of saturation to edge-ordered graphs. However, when adding a non-edge, now there are several choices on how we can add it to the order of the edges of H. Unlike for vertex-ordered graphs that were studied in <cit.>, this is not uniquely defined. Thus, we define three variants:
	
	 
		We are given an edge-ordered graph G.
		
		Let sat_e(n,G) be the minimum number of edges in an edge-ordered graph H on n vertices in which adding any new edge with any new label (inserted anywhere in the linear order of edges) introduces a copy of G (that is, every label is forbidden; e stands for `every' or `edge-ordered').
		
		Let sat_m(n,G) be the minimum number of edges in an edge-ordered graph H on n vertices in which adding any new edge with a minimal label (inserted at the beginning of the linear order of edges) introduces a copy of G (m stands for `minimal').
		
		Let sat_s(n,G) be the minimum number of edges in an edge-ordered graph H on n vertices in which any new edge can be labeled by some single label (inserted somewhere in the linear order of edges) such that this introduces a copy of G
		(s stands for `some' or `single').
		
		In each case a (not necessarily minimal) H with the above saturation property is called a host graph and we say that H saturates G.[It will be always clear from the context which definition we are talking about.]
	
	
		From the definition we see that if H is a saturation host graph for sat_e, then H  is a saturation host graph also for sat_m.  Similarly, if H is a host graph for sat_m, then it is a host graph also for sat_s. This implies the following inequalities:
	
	
		For every edge-ordered graph G, we have 
    sat_s(n,G) ≤ sat_m(n, G) ≤ sat_e(n, G)≤ ex_e(n, G).

	
	

	Therefore, the most natural saturation function is sat_e(n,G), which we get from the extremal problem by searching for a minimal saturating host graph instead of a maximal one.[This is why e refers to `edge-ordered' alongside `every'.] However, it turns out that in many ways instead of dealing with every possible label of the new edge, we get the same behavior already if we forbid the minimal and the maximal label. In fact, we get a lot of insight already by looking at the minimal label only, which is slightly more convenient to handle. Therefore, our center of attention is sat_m through most of the paper. The consequences for sat_e are discussed only afterwards, while we consider sat_s only briefly compared to the other two functions. 
	
		
	
	
	
	We also consider a common variant of the saturation problem, when we do not assume that a graph H is G-free. Therefore, we only ask for a graph with minimum number of edges such that, by adding a new edge with every/minimal/some label, a new copy of G is created. This version is called semisaturation and the respective functions are denoted by ssat_e/ssat_m/ssat_s. By definition, the following holds, which will be used throughout the paper.
	
	
		For every edge-ordered graph G, we have 
    ssat_e(n,G) ≤ sat_e(n, G),    
    		ssat_m(n,G) ≤ sat_m(n, G),    
    		ssat_s(n,G) ≤ sat_s(n, G).

	
	
	

 §.§ History

	
	Saturation problems were first studied by Erdős, Hajnal and Moon <cit.> in 1964 as an analogue of the extremal problem. In 1986, Kászonyi and Tuza <cit.> characterized saturation functions for graphs by showing that a graph G has bounded saturation function if and only if G has an isolated edge, otherwise its saturation function is linear. Since then, saturation problems became an important topic in extremal combinatorics. For an updated survey, see <cit.>. Problems regarding saturation appear in the literature in many different variants and for diverse combinatorial objects. During the last few years, a significant progress was made in studying saturation of 0-1 matrices, or equivalently vertex-ordered bipartite graphs. Brualdi and Cao <cit.> were first to investigate their saturation functions. Shortly after that, Fulek and Keszegh <cit.> managed to show that dichotomy holds for 0-1 matrices too, identifying large families of matrices with linear saturation functions. On the other hand, Geneson <cit.> found an infinite family of matrices with bounded saturation function. Finally, Berendsohn <cit.> gave a complete characterization of permutation matrices with bounded saturation function. 
	
	Inspired by these interesting results about 0-1 matrices, the authors <cit.> initiated the study of saturation problems on vertex-ordered graphs. They considered two different orders on vertex sets, linear and cyclic order. They proved dichotomy in both cases and they found infinite families of graphs that have bounded (respectively linear) saturation functions. However, even for perfect matching graphs (which are the natural counterparts of permutation matrices from the 0-1 matrix setting), the complete characterization is still unknown. 
	
	On the other hand, saturation problems for directed graphs were not yet studied systematically. Füredi et al. <cit.> showed in a different context that C_3, a directed cycle of length three, has a superlinear semisaturation function. More precisely, ssat(n, C_3) = n log n + O(n loglog n). This implies that the saturation function is also superlinear. In <cit.> they ask for a matching upper bound for sat(n, C_3). In fact leaving out the two special vertices from the construction of <cit.> shows that sat(n, C_3) = n log n + O(n loglog n) as well, answering this question.[This construction saturating C_3 is the following. Let k be an appropriate even number. We take a bipartite graph with parts of size 2k and at most 2kk for an appropriate k, on the right side the vertices correspond to different sets of size k of the left side and then we orient an edge from w towards v_E corresponding to some set E if and only if w∈ E. It is easy to check that this graph avoids C_3, as even without the ordering it does not contain a triangle. Also, adding any new edge with any orientation introduces a copy of C_3.] This was observed by Alon and Fox <cit.>. 
		
	
	Pikhurko <cit.> considered a saturation problem on the class of cycle-free directed graphs, i.e. adding any missing edge either introduces a cycle or a copy of a directed graph G, and showed that any family ℱ of cycle-free directed graphs has sat(n, ℱ) = O(n) with respect to this definition.
	
	The study of extremal problems of edge-ordered graphs leads also to interesting problems. Their systematic study was started by Gerbner et al. <cit.>. They proved an Erdős-Stone-Simonovits type theorem which gives the exact asymptotic (a quadratic function) for the extremal function if the so-called order chromatic number of the forbidden graph is at least 3. Thus, graphs with order chromatic number 2 are the most interesting to study further. They systematically took account of edge-ordered paths with 4 edges.
	This study was continued by Kucheriya and Tardos <cit.>, <cit.>. They characterized connected edge-ordered graphs that have a linear extremal function and showed that for every other connected edge-ordered graph the extremal function is Ω(nlog n), a dichotomy phenomenon similar to what happens for vertex-ordered graphs. They also showed that for edge-ordered forests of order chromatic number 2 the extremal function is n· 2^O(√(log n)). Note that if an edge-ordered graph contains a cycle then its extremal function is Ω(n^c) for some c>1, which is implied by the corresponding unordered graph result.
	
	The motivation to study extremal problems of vertex-ordered graphs came from combinatorial geometry where several natural problems can be described in such a way. It turns out that edge-ordered graphs have similar applications, as shown already in <cit.>, and later in <cit.>.
	
	Continuing this line of research, here we initiate the study of saturation problems of edge-ordered graphs.
	
	

 §.§ Main results

	
	For (unordered) graphs <cit.>, 0-1 matrices <cit.> and vertex-ordered graphs <cit.> it was shown that the saturation functions are either O(1) or Θ(n). 	
	Our main and somewhat surprising result is that there is no such dichotomy for edge-ordered graphs, similar to directed graphs. We first discuss sat_m definition, then the remaining two.
	
	The diamond graph is the graph we get by removing an edge from K_4.
	We consider the diamond graph with different edge-orderings. We prove that for certain edge-orders, the saturation function is neither O(1) nor Θ(n). More specifically, we use a result by Katona and Szemerédi <cit.> about bipartite coverings to establish a lower bound Ω (n √(log n)). 
	
	Furthermore, we generalize the lower bound obtained for diamond graphs to an infinite family of edge-ordered graphs. On the other hand, we prove that at least for one of these diamond graphs the saturation function is at most O(n log n/loglog n). Additionally, we find an infinite family for which this upper bound holds. However, we do not know whether even a weaker non-trivial upper bound can be shown for arbitrary edge-ordered graphs. 
	
	We also find infinite families of graphs with bounded and linear saturation functions. Among others, if T is an edge-ordered tree, then sat_m of e_0 + T is bounded, where e_0+T denotes the graph we get from T by adding an isolated edge with minimal label. From now on e_0 (resp. e_max) always refers to the edge with minimal (resp. maximal) label and addition of graphs refers to their disjoint union.
		
	Altogether, our examples show that the saturation functions can be bounded, linear and superlinear. In fact we show infinitely many examples for each case.
	
	However, dichotomy holds in a weaker sense for sat_m. We can show that if we assume that the minimal edge e_0 is isolated, then sat_m is either bounded or linear (it is even possible but we could not prove it, that the latter case never happens). Otherwise, if the minimal edge is not isolated, then sat_m is at least linear. 
	
 	A summary of our saturation results for sat_m definition is shown in Table <ref>.
		
	
	
	Next we concentrate on the semisaturation problem of edge-ordered graphs. As opposed to 0-1 matrices <cit.> and vertex-ordered graphs <cit.> where it was relatively simple to fully characterize semisaturation functions, for edge-ordered graphs it appears to be more delicate. It turns out that the examples with superlinear saturation functions also have superlinear semisaturation functions. Our main result about the semisaturation functions is a general upper bound O(n log n). We are also able to characterize the edge-ordered graphs with bounded ssat_m, they are the ones with isolated e_0. Further, we identify a large family of graphs with linear semisaturation functions. Altogether, for semisaturation functions we also show infinitely many examples for each case (bounded, linear, superlinear).
	A summary of our semisaturation results for ssat_m definition is shown in Table <ref>.
	
	

	We further study the other two definitions that we proposed: sat_e and sat_s. We show several results following the proofs for sat_m definition. First of all, we show a weak dichotomy for sat_e, then we characterize graphs with bounded semisaturation function ssat_e. Moreover, we notice that the general semisaturation bound also holds for this definition. Later on, we find infinite families with bounded, linear and superlinear (semi)saturation function. We study the family of edge-ordered graphs F_k = e_0 + G_k + e_max, where e_0 is the minimal edge and e_max is the maximal edge of F_k, and the underlying graph of G_k is a complete graph K_k. It turns out that in this case, F_k can have a superlinear sat_e function for certain labelings of G_k. This result shows how differently sat_e and sat_m functions behave, since for sat_m we are able to show that sat_m(n, e_0 + G) = O(n) for any edge-ordered graph G. We also show that cycles of odd length have linear saturation and semisaturation functions and any edge-ordered matching has bounded saturation function. The summary of the results about sat_e and ssat_e functions can be seen on Table <ref>.
	
	Considering sat_s, note that by definition most of the results about sat_s are more general than for sat_m, because any condition that is imposed on the minimal edge e_0 can be replaced by an equivalent condition on an arbitrary edge in a given graph. Nevertheless,	
	all our results about sat_s follow as corollaries of some results about sat_m functions. 
	
	
	
	

§ DICHOTOMIES AND GENERAL UPPER BOUNDS

	
	

 §.§ sat_m

	
	In the vertex-ordered setting it was possible to show that the saturation function of any graph is either bounded or linear <cit.>. Such a dichotomy does not hold for edge-ordered graphs in general, as we will see later. However, we can still separate between bounded and at least linear saturation functions. In addition, if the minimal edge is isolated, then we do have a dichotomy. Note that the isolated edge case was the only interesting case for vertex-ordered graphs <cit.>, as the remaining vertex-ordered graphs all have linear saturation function.
	
	 
		Let G be an edge-ordered graph. If the minimal edge e_0 of G is isolated, then either sat_m(n,G) = O(1) or sat_m(n,G) = Θ(n). Otherwise, if e_0 is not isolated, then sat_m(n,G) = Ω(n).
	
	
	
		Assume first that e_0 is not isolated. There cannot be two isolated vertices in a saturating host graph H, otherwise by connecting them we will not introduce a new copy of G. Thus, sat_m(n,G) ≥ n/2-1 and the result follows.
		
		Now, we assume that e_0 is isolated. To prove that sat_m(n,G) = O(n) for any G it is enough to take H to be the union of G - e_0 and n-k+2 isolated vertices, where k is the number of vertices of G. If we add an edge between any two isolated vertices, a copy of G will be obtained. Next, we can create a saturating host graph H' by greedily adding edges to H (a common trick for other structures as well, e.g., for vertex-ordered graphs), such that H' can have at most n2 - n-k+22≤ kn edges. This implies that sat_m(n,G) = O(n). On the other hand, if G has sublinear saturation number, then for every large enough n_0 there is a host graph H_0 on n_0 vertices that has two isolated vertices. Then it is easy to see that by adding isolated vertices to H_0 we still must have a saturating host graph. Therefore, we conclude that if the saturation number is not linear, then it has to be bounded.
	
	
	It is actually possible that if e_0 is isolated, then sat_m is always bounded. We can prove this in the case of semisaturation.
	
	 
		Let G be an edge-ordered graph. If the minimal edge e_0 of G is isolated, then ssat_m(n,G) = O(1), otherwise ssat_m(n,G) = Ω (n). 
	
	
		Assume that e_0 is not isolated. If there are two isolated vertices in the host graph H, by connecting them we do not introduce a new copy of G. Thus, sat_m(n,G) ≥ n/2-1. This is the same proof as in Theorem <ref>.
		
		Assume now that e_0 is isolated. Take H to be a graph on n vertices that has two disjoint copies of G and all the remaining vertices isolated. Then we add edges between non-isolated vertices to get a host graph with a bounded number of edges.
	
	
	We now proceed to prove a general semisaturation upper bound. We denote by N_G(a) (resp. N_G[a]) the open (resp. closed) neighborhood of a in G.
		
	 
		For every edge-ordered graph G, we have ssat_m(n,G) = O(n log n).
	
	
	
		Let G be labeled with a function l: E(G) →{0, 1, ..., m} and denote by e_0 = ab its minimal edge. We define an additional labeling function for host graphs L: E(H) →ℕ^3, where ℕ^3 is equipped with lexicographic order. We write x.y.z for (x,y,z) ∈ℕ^3, and x.y for (x,y,0). Let N_G(a) ∖ b := {a_1, …, a_r} and N_G(b) ∖ a := {b_1, …, b_s}. Let  k = ⌈log n ⌉ and n'< n determined later. Consider k disjoint copies of the graph G^ab = G ∖{a,b}, which we denote by G^ab_i, for 1 ≤ i ≤ k. For every edge e ∈ E(G^ab), we label the corresponding edge e_i ∈ E(G^ab_i) by: 
    L(e_i) = l(e).i,  for all  1 ≤ i ≤ k.
 Furthermore, we add n' isolated vertices V = {v_1, ..., v_n'}. 
		
		Take k bipartitions (X_i,Y_i) of V such that every pair of vertices is separated in one of them (this can easily be done as n'≤ 2^k, e.g., the ith bipartition is according to ith digit in the binary form of the indices of the vertices).
		
		Fix some i≤ k. Then, take all the vertices a_i,j in G^ab_i that correspond to the neighbors of a. We add an edge between v_t ∈ X_i and a_i,j, where j ∈{1,...,r}. We label such edges with the following function:
		
    L(v_ta_i,j) = l(aa_j).i.t,  for all  1 ≤ j ≤ r.

		
		Analogously, we do the same for the neighbors of b, but this time by connecting them with isolated vertices v_t in Y_i. We label them by:
		
    L(v_tb_i,j) = l(bb_j).i.t,  for all  1 ≤ j ≤ s.

		
		This graph has O(kn) edges.
		
		We can see that by adding an edge between a vertex x_i ∈ X_i and a vertex y_i ∈ Y_i, the graph induced by G_i^ab and these two vertices is isomorphic to G, where the edge x_iy_i plays the role of the minimal edge ab. 
		
		As every pair of vertices in V is separated in some bipartition, adding an edge between any two vertices in V creates a copy of G.
		
		Set n' such that this graph has exactly n vertices. By greedily adding edges between vertices that are not in V, we obtain a host graph that semisaturates G and has O(n log n) edges.	
	
	
	While the upper bound O(n log n) could also be true for sat_m, we can prove it only for a subclass of graphs.
	
	 
		Let G be an edge-ordered graph such that G-e_0 is bipartite and N_G[a] = N_G[b], where e_0 = ab is the minimal edge. Then sat_m(n,G) = O(n log n).
	
	
	
		We construct the graph H the same way as in Theorem <ref> and we show that H is G-free. First, observe that if N_G[a] = N_G[b] = {a,b}, then e_0 is an isolated edge and the result follows by Claim <ref>. Otherwise, there exists a vertex w that creates a triangle with a and b, which means that G is non-bipartite. We will show that H given in Theorem <ref> is bipartite and it follows that H avoids G. Then we again greedily extend H to get a saturating host graph for G.
				
		Let (X, Y) be a bipartition of G-e_0 (which exists by the assumption of the theorem), then notice that a and b should be in the same part, say X, while all of their neighbors N^ab := N_G(a) ∩ N_G(b) have to be in Y.
		Since G-e_0 is bipartite, then G_i^ab is also bipartite for every 1 ≤ i ≤ k. We now can construct a bipartition (X_H, Y_H) of H, and therefore prove the claim. We take a disjoint union of all G_i^ab, such that the corresponding neighbors N_i^ab of a and b are all in Y_H. Then we add the set V as defined in the proof of Theorem <ref>, to the set X_H and we connect all the vertices in V with all the vertices in ⋃_i=1^k N_i^ab, which gives a desired bipartition of H. So, we can conclude that H is indeed G-free.
	
	
	

 §.§ sat_e

	We are only able to prove a weak dichotomy for sat_e. We denote by e_0 the minimal edge and by e_max the maximal edge of G.
	
	 
		Let G be an edge-ordered graph. If e_0 and e_max are isolated, then either sat_e(n,G) = O(1) or sat_e(n,G) = Ω(n). Otherwise, sat_e(n,G) = Ω(n).
	
	
		We follow the same ideas as in the proof of Theorem <ref>. First we consider the case when either e_0 or e_max is not isolated. Assume wlog. e_0 is not isolated, then if we label one missing edge with a minimal label between two isolated vertices of H, no copy of G is created. Therefore, H does not have two isolated vertices and we get sat_e(n,G) ≥ n/2-1. 
		
		Let e_0 and e_max be isolated, and assume that G has a sublinear saturation function. Then for a large enough n, the host graph must have at least two isolated vertices. Since they can be replaced by arbitrary many isolated vertices, without affecting saturation property, we conclude that if the saturation function is sublinear, then it has to be bounded.		
	
	
	Notice that in the e_0 and e_max isolated case when sat_e is unbounded, unlike in Theorem <ref> where we proved Θ(n), here we only have Ω(n). The reason for this is detailed later.
	
	Similarly to Claim <ref>, for semisaturation we can prove a stronger statement.
		
	 
		Let G be an edge-ordered graph. If e_0 and e_max are isolated, then ssat_e(n,G) = O(1), otherwise ssat_e(n,G)=Ω (n).
	
	
		Let H be a host graph on n vertices and m edges, such that H = e_1 + G_1 + G_2 + G_3 + G_4 + e_m + V where G_i are copies of G - {e_0, e_max} so that l(e_1) < l(G_1) < l(G_2) < l(G_3) < l(G_4) < l(e_m) and V, the set of the remaining vertices of H, is a set of isolated vertices.  Let e'=uv be an edge that we can add to H with u∈ V isolated. We distinguish two cases:
		
			
  * l(e') > l(G_2). If v ∈ V(G_2 + G_3 + G_4 + e_m), then e_1 + G_1 + e' is a copy of G. Otherwise, e” + G_2 + e' where e”∈ E(e_1 + G_1). If G_1 has no edges, then e' + G_2 + e_m is a copy of G.
			
  * l(e') < l(G_3). Similarly, if v ∈ V(e_1 + G_1 + G_2 + G_3), then e' + G_4 + e_m is a copy of G. Otherwise, e' + G_3 + e” where e”∈ E(G_4+e_m). If G_4 has no edges, then e_1 + G_3 + e' is a copy of G.
		
		Thus, adding greedily edges to H that are not in V, we get a host graph that semisaturates G with O(1) edges.
		
		Otherwise, if for example e_0 is not isolated, then by adding any new edge with minimal label between two vertices in V we do not create a copy of G. Therefore, the semisaturation function is at least linear.
	
	
	We are also able to show the general upper bound for semisaturation. Let G^rev be the edge-ordered graph obtained from G such that the underlying graph remains the same, but the linear order of its edges is reversed. By Theorem <ref> we get that sat_m(n,G) = O(n log n) and sat_m(n,G^rev) = O(n log n). By gluing the host graphs H and H^rev on the same independent set V such that l(H^rev) < l(H), we get a new host graph which semisaturates G in sat_e sense for any missing edge in V.
	
	 
		For every edge-ordered graph G, we have ssat_e(n,G) = O(n log n).
	

	Unlike for sat_m definition, we do not know whether it is possible to show this upper bound for saturation function of some family similar to the one described in Theorem <ref>.
	
	

 §.§ sat_s

	
	Recall that sat_s(n, G) ≤ sat_m(n, G) for every edge-ordered graph G. So, all the results about upper bounds can be applied to sat_s definition. In this subsection, we list the main corollaries. 
	The first result is the analogue of Theorem <ref>.
	
	 
		Let G be an edge-ordered graph. If G contains an isolated edge, then either sat_s(n,G) = O(1) or sat_s(n,G) = Θ(n). Otherwise, sat_s(n,G) = Ω(n).
	
	
	We continue with the weak dichotomy of semisaturation functions. The proof is completely analogous to the proof of Claim <ref>.
	
	 
		Let G be an edge-ordered graph. If G contains an isolated edge, then ssat_s(n,G) = O(1), otherwise ssat_s(n,G)=Ω (n).
	
	
	The general upper bound for semisaturation follows directly from Theorem <ref>.
	
	 
		For every edge-ordered graph G, we have ssat_s(n,G) = O(n log n).
	

	We conclude by defining a family of graphs for which this upper holds for the sat_s function. 
	
	 
		Let G be an edge-ordered graph. If there exists an edge ab ∈ E(G) such that G-ab is bipartite and N_G[a] = N_G[b]. Then sat_s(n,G) = O(n log n).
	

	Note that these results are more general than for sat_m definition, which comes from the fact that we can usually replace the assumptions about the minimal edge e_0 to any edge in G. In Section <ref>, we will get back to this definition and explain several other results.
	
	

§ UNBOUNDED SATURATION FUNCTIONS

	
	In this section we deal only with sat_m and ssat_m, the other variations are discussed in later sections.
	
	

 §.§ Bipartite coverings

	
	We will need an auxiliary result about bipartite coverings of almost complete graphs. 
	
	A bipartite covering of a graph G is a collection of bipartite graphs, so
	that each edge of G belongs to at least one of them. The capacity of the covering is the sum of the numbers of vertices of these bipartite graphs.
	We need a lower bound on the capacity of a bipartite covering of almost complete graphs. 
	
	<cit.>
		Let G be a graph on n vertices and let d_1, d_2, . . . , d_n denote the degrees of its vertices. Then the capacity of any bipartite covering of G is at least
		
    ∑_i=1^n(log n- log (n-d_i)).

	
	
	This was shown by Katona and Szemerédi <cit.>, see also the paper of Alon <cit.> for a generalization which also has a similar and short proof. See also the exact result of Dong and Liu <cit.> that in a bipartite covering of the complete graph K_n there is a vertex which has incident edges from at least log n of the bipartite graphs.
	
	
		Let G be a graph with at most cn^2-ϵ missing edges for some constants c,ϵ > 0 and let B be a bipartite covering of G, then there is a vertex which has incident edges from c' log n of the bipartite graphs in B, where c' depends only on c and ϵ.		
	
	
	
		Let x=2cn^1-ϵ. We delete at most n/2 vertices from G that have co-degree at least x, and denote by G' the obtained graph. Then G' has n'≥ n/2 vertices and their co-degrees are less than x and so their degrees are more than n'-x-1. 
		
		By Theorem <ref>, the capacity of B restricted to G' is at least 
    ∑_i=1^n'(log n'- log (n'-d_i))> n/2(log n'- log (x+1))= n/2(ϵlog n-O(1)),

		
		where the O notation hides dependence on c and ϵ. Therefore, one of the vertices must be in at least c'log n bipartite graphs for some constant c'.
	
	
	

 §.§ Superlinear saturation functions

		
	Later we will see examples of graphs with bounded saturation function. Next we show a family of graphs that have superlinear saturation and semisaturation functions, showing that there is no dichotomy. 
	
	Notice that, by Theorem <ref> dichotomy does not hold neither for saturation nor for semisaturation. This most likely suggests that studying semisaturation of edge-ordered graphs might be more demanding than it was for vertex ordered graphs <cit.>, where it was possible to fully  characterize semisaturation functions.
	
	 
		Let G be an edge-ordered graph such that its minimal edge is ab. If there exist two vertices v,w ∈ N_G(a) ∩ N_G(b) such that l(av) < l(aw) and l(bv) > l(bw), then sat_m(n,G) ≥ ssat_m(n,G)=Ω(n√(log n)).
	
	
	
		Let H be a saturation host graph on n vertices. For a pair of vertices that do not span an edge of H, denoted by f, adding an edge at f with minimal label we should get a copy of G. Thus there exists a pair of vertices e=π(f) such that the four vertices f∪ e span a copy of the structure on four vertices and five edges which is required by our assumption in which f plays the role of ab. Note that the order of labels of the two edges going from the two vertices of f to the two vertices of e is reversed. Thus, for every pair of vertices e there is a bipartition of the vertices such that every non-edge in the set π^-1(e) goes between the two parts of this bipartition.
		
		Assume on the contrary that H has less than cn√(log n) edges for some c≤ 1 chosen later. The non-edges of H must be covered by the sets of non-edges π^-1(e) where e is a pair of vertices in H. As each π^-1(e) is bipartite, we can apply Corollary <ref> on the complement of H.	First we choose constants c_0,ϵ_0 independently from c such that n√(log n)≤ c_0(n/2)^2-ϵ_0 holds. Now we can apply Corollary <ref> with c_0,ϵ_0.		
		We get that there is a vertex v in H which participates in at least c'log n many bipartitions that cover the non-edges of H. For each 
		e such that π^-1(e) contains a non-edge that covers v, there is an edge in H from v to both vertices of e. Thus v is connected to at least c'log n different such pairs of vertices, therefore, to at least √(c'log n) vertices in H. Delete v from H. Repeat this procedure n/2 times, in each step we can apply Corollary <ref> with the same c_0 and ϵ_0 to find and delete a vertex with degree at least √(c'log (n/2)). Altogether, we have found at least n/2√(c'log n) edges in H. When c is small enough this contradicts our assumption that H has less than cn√(log n) edges, .
	
	
	
	
	 
		Let D_0, D_1, D_2 be the graphs shown in Figure <ref>. Then ssat_m(n,D_i)=Ω(n√(log n)), for i ∈{0,1,2}. 
	

	We would like now to understand better which edge-ordered graphs are not covered by Theorem <ref>. We will discuss later the graphs where the endvertices of the minimal edge have disjoint neighborhoods. For now, we assume that N_G(a) ∩ N_G(b) is non-empty.

	 
		Let {u_1, u_2, ..., u_k} := N_G(a) ∩ N_G(b), for k ≥ 1. We can assume wlog. that l(au_1) < ... < l(au_k). If l(bu_1) < ... < l(bu_k) does not hold, then there exist 1 ≤ i < j ≤ k such that l(au_i) < l(au_j) and l(bu_i) > l(bu_j), which by Theorem <ref> implies that G has a superlinear saturation function.
	
	
	We now proceed by establishing an upper bound for D_0 which is close to its lower bound.
	
	 
		
    ssat_m(n,D_0) ≤ sat_m(n,D_0)=O(n log n/loglog n).

	
	
	
		We show the upper bound by constructing a certain labeling of the complete bipartite graph H = K_k, n (for appropriate small k) which will be a host graph for D_0. Since D_0 is non-bipartite, it follows that H is D_0-free regardless of its labels. We will label the edges of H such that it saturates D_0 for the non-edges within the K̅_n part, at the end we just greedily extend this graph with edges put within the K̅_k part to get a host graph for D_0. This time, we assume that the labeling function takes values in ℝ_+. 
		
		An alternative way to look at this problem is to describe the labeling in terms of the biadjacency matrix of H that we denote by L = (ℓ_i,j)_1 ≤ i ≤ k, 1 ≤ j ≤ n with entries from ℝ_+. We say that a matrix L satisfies the sat property if the following holds:
		for every two column indices 1 ≤ j_1 , j_2 ≤ n, there exist two row indices 1 ≤ i_1 , i_2 ≤ k such that we have ℓ_i_2,j_1 < ℓ_i_1,j_1 < ℓ_i_1,j_2< ℓ_i_2,j_2. It is easy to see that if we label the edges of H according to an L that satisfies the sat property, then we get a host graph that saturates D_0 for the non-edges within the K̅_n part of H (the sat property is enough but not necessary for that). Notice that L has the sat property if and only if for every pair of columns the subgraph defined by these two columns has the sat property. See Figure <ref> for an illustration.
		
		
		
		
		First, let us assume that n=k! for some k ∈ℕ and show that the matrix L of size k× n which will be determined below satisfies the sat property. We partition the elements of the matrix L into blocks (B_i,j)_1 ≤ i ≤ k, 
 1 ≤ j ≤ i! such that the ith row is equipartitioned into i! blocks:
		
    B_i,j = {ℓ_i,(j-1)n/i!+1, … , ℓ_i,jn/i!}.

		
		Considering how the blocks in each row partition the columns, we can observe this partition for any given row is a refinement compared to the row above, that is, for each block B in the (i-1)th row there are exactly i blocks in the ith row such that their columns equipartition the columns of B. See Figure <ref> for an example.
		
		The labeling will be such that within any block the labels are in increasing order, that is, ℓ_i,j_1 < ℓ_i,j_2 for any two entries in the same block whenever j_1 < j_2. Additionally, for any two blocks their labels can be separated, that is, for any two blocks B_i_1, j_1, B_i_2, j_2 all the elements of B_i_1, j_1 are smaller (or larger) than all the elements of B_i_2, j_2. We denote this by  B_i_1, j_1 < B_i_2, j_2 (or B_i_1, j_1 > B_i_2, j_2). 
		
		The labeling is defined row by row. First, we label the entries in B_1,1, the first row of the matrix, with any n different positive real numbers in increasing order. Then we proceed to the second row, by labeling B_2,1 and B_2,2 such that B_2,1 < B_1,1 < B_2,2. Note that, if we take any two columns j_1 and j_2 such that j_1 ≤n/2 < j_2, then these two columns and the first two rows of the matrix L define a submatrix that has the sat property. We continue with the third row by labeling the blocks such that
		
    B_3,1 < B_2,1 < B_3,2 < B_1,1 < B_3,3 and  B_3,4 < B_1,1 < B_3,5 < B_2,2 < B_3,6.

		See again Figure <ref>, for an example when n = 6 and observe that it satisfies the sat property.
		
		
		
		In a general step, if we assume that i-1 rows are labeled for i ≥ 2, then we label the ith row as follows. We take any j such that 1 ≤ j ≤ (i-1)! and we want to label all the blocks in row i that share a column with B_i-1,j. These are exactly the blocks B_i,(j-1)i+1, …, B_i,ji and we label them such that
		
    B_i,(j-1)i+1 < A_1 < B_i,(j-1)i+2 < ... < A_i-1 < B_i,ji,

		where {A_1, …, A_i-1} are the blocks that share a column with B_i-1,j (including B_i-1,j), (as there is exactly one such block in each row). Additionally, these are indexed in an increasing order, i.e., A_1 < … < A_i-1. In the example above, if we want to label the block B_3,2, then A_1 := B_2,1, A_2 := B_1,1.
		
		Next we prove by induction on i that if we take any two columns j_1 and j_2 that correspond to different blocks in the ith row, then these two columns and the first i rows of the matrix L define a submatrix that has the sat property. This holds for i=2. Suppose i>2 and the two columns correspond to different blocks of the row i-1, then this holds by induction. Finally, if i>2 and the two columns were separated first in the ith column, say they are in blocks B_1 and B_2, where B_1 comes first (has smaller column indices). By the definition of labeling, we have B_1<B_2. Also, in each earlier row there is a block that contains both columns and there is one such block A in some row i' for which B_1<A<B_2. As within A the entries are in the increasing order, the four entries in these two columns and in rows i,i' show that the submatrix indeed has the sat property.
		
		When we reach the last row, we get that the blocks are all singletons and we stop. As each column is separated in this last row, by the inductional claim each pair of columns define a submatrix with the sat property and thus the whole matrix L also has the sat property, as required.
		
		To describe k in terms of n we can use Stirling's formula to obtain k log k = Θ(log n), which is equivalent to
		
    c_1 log n/log k≤ k ≤c_2 log n/log k,

		for some positive constants c_1 and c_2. From the first inequality we can get that c_1 log n ≤ k log k ≤ k^2 which is equivalent to √(c_1 log n)≤ k. If we plug this into the second inequality we get
		
    k ≤c_2 log n/log√(c_1 log n) = O(log n/loglog n).

		
		Thus we get a labeling of H from the matrix L. Finally, we greedily keep adding edges with minimal labels to H within the K_k part until the graph saturates D_0. As the number of edges in H is at most kn+k 2, the theorem follows.
		
		We now handle the case when there is no k ∈ℕ such that n=k!. In this case set k to be the smallest number with n<k! and determine the k× n matrix L the same way except that when defining the blocks in the ith row, instead of equipartitioning the columns of each block in the (i-1)th row,  we now partition them to i almost equal size blocks (i.e., the difference of their sizes is at most one). Note that since (k-1)! < n, then in (k-1)th row, each block is of size at least one. Only in the last row we may have blocks of size zero, but this does not change asymptotically the result and it is straightforward to check that the same proof works.
	
	
	Even though the bounds are not far, determining the exact order of magnitude of sat_m(n,D_0) is left open. We can generalize this upper bound for further graphs. In order to state this result, we introduce the following notation: for two edge-ordered graphs (usually disjoint subgraphs of some graph) we write l(F_1) < l(F_2) if all the labels of F_1 are smaller than all the labels of F_2.
	
	
		Let F_1 and F_2 be two edge-ordered graphs such that sat_m(n, F_1) = O(f(n)), where f(n) = Ω(n) and l(F_1) < l(F_2). Then sat_m(n, F_1 + F_2) = O(f(n)).
	
	
		Let H(F_1) be the host graph for F_1, and assume F_2 has k vertices. Then we take a union H(F_1) + F_2^*, where F_2^* is a relabeled copy of F_2 such that l(H(F_1)) < l(F_2^*). We argue that this construction indeed avoids F_1+F_2. Assume there is a copy of F_1 in H(F_1) + F_2^*, clearly it cannot be found in the part H(F_1), so it has to have at least one edge in F_2^*. In that case F_2 has to be found in the rest of F_2^*, which is not possible because of its size. 
		
		Notice that by adding any edge within H(F_1) we get a copy of F_1+F_2, and we can greedily add any other edge if it does not create a copy of F_1+F_2, which only adds at most kn+k^2 edges. Since f(n) is at least linear, we get that sat_m(n, F_1 + F_2) = O(f(n)).
	

	Using this result, we can generalize the upper bound obtained for D_0 to an infinite family of edge-ordered graphs.
	
	 
		Let G be an edge-ordered graph, such that l(D_0) < l(G). Then 
    ssat_m(n,D_0+G)≤ sat_m(n,D_0+G) = O(n log n/loglog n).

	

	

 §.§ Linear saturation functions

	
	We discuss now the remaining three graphs that have the diamond graph as their underlying graph.
	
	 
		For i∈{3,4,5} we have sat_m(n, D_i)= Θ(n), where D_i are the graphs as shown in Figure <ref>.
	
	
	
		By Theorem <ref>, it is enough to show that sat_m(n, D_i)= O(n) for all i∈{3,4,5}. Take H to be a host graph on n vertices such that its underlying graph is a complete bipartite graph K_2,n-2. Denote the vertices of H by u_1, u_2 on one side and v_1, ..., v_n-2 on the other side. With a proper edge-ordering we will be able to define for each D_i a saturating graph H_i that has linearly many edges, from where the conclusion will follow. Notice also that since in all the cases the underlying graph of H_i is bipartite, and D_i is not, we can easily conclude that H_i is D_i-free. Therefore, we will only need to check that H_i actually saturates D_i.
		
		We start with D_3, the following labeling of H gives H_3:
		
    l(u_1v_i) = 2i-1  and  l(u_2v_i) = 2i , for all  i ∈{1,2, ..., n-2}.

		To check that H_3 saturates D_3, connect an edge v_iv_j for any 1 ≤ i < j ≤ n-2 and label it with zero. Then, notice that the graph induced by four vertices u_1, u_2, v_i, v_j is isomorphic to D_3. We can add the edge u_1u_2 for example with the largest label, this does not affect the linearity of sat_m(n, D_3).
		
		We continue with the case D_4, the following labeling of H gives H_4:
		
    l(u_1v_i) = i  and  l(u_2v_i) = n-2+i , for all  i ∈{1,2, ..., n-2}.

		Again, as in the previous case H_4 saturates D_4, as after adding an edge between v_i and v_j labeled with zero, the graph induced by these two vertices and u_1 and u_2 is isomorphic to D_4. We can add the edge u_1u_2 with the largest label, this does not affect the linearity of sat_m(n, D_4). Thus, we are done.
		
		Finally, we treat the case D_5, the following labeling of H gives H_5:
		
    l(u_1v_i) = i  and  l(u_2v_i) = 2n-3-i , for all  i ∈{1,2, ..., n-2}.

		This can be checked analogously to the previous two cases.
	
		
	
	
	 
		Let G be an edge-ordered graph and let ab be the minimal edge in G such that deg(a) ≥ 2. If N_G(a) ∩ N_G(b) = ∅, then ssat_m(n,G) = Θ(n).
	
	
	
		The lower bound follows from Claim <ref>. For the upper bound let N_G(a) ∖{b} ={u_1, u_2, ..., u_s} and N_G(b) ∖{a} = {v_1, v_2, ..., v_t}, and assume that G has r vertices. Furthermore, let H' be the graph obtained by taking the union of G ∖{a, b} and K_n-r+2 (the complement of the complete graph, that is, an independent set), whose set of vertices we denote by {w_1, w_2, ..., w_n-r+2}. Then, the host graph H on n vertices is obtained by adding edges u_iw_k and v_jw_k to H', for i ∈ [s], j ∈ [t], and k ∈ [n-r+2]. We label them as follows: L(u_iw_k) = l(u_ia).k, and L(v_jw_k) = l(v_jb).k, while all the remaining edges keep the same labels as in G.
		
		Notice that H semisaturates G for the edges inside K_n-r+2. Indeed, if we add any edge of the form w_iw_j for i,j ∈ [n-r+2], it will have a role of the minimal edge ab in the new copy of G. 
		
		We can greedily add a linear number of edges incident to the part G ∖{u, v} to get a semisaturating host graph.
	

	A natural family that satisfies these conditions is the family of triangle-free graphs (such that e_0 is not isolated).

	 
		Let G be a triangle-free edge-ordered graph, whose minimal edge is not isolated. Then ssat_m(n, G) = Θ(n).
	
	
	 
		Let G be a non-bipartite edge-ordered graph. Let e_0 = ab ∈ E(G) be the minimal edge and assume that G-e_0 is bipartite and N_G(a) ∩ N_G(b) = ∅, then sat_m(n,G) = Θ(n).
	

	
		We use the same construction as in Theorem <ref>. To verify that the host graph is G-free, notice that we obtain the host graph by taking the bipartite graph G - ab and replacing the vertices a and b by arbitrary many vertices which are connected to N_G(a) ∪ N_G(b). This means that the host graph H remains bipartite and consequently it cannot contain a copy of non-bipartite graph G.
	
	
	From this statement, we get that the cycles of odd degree have linear saturation function. We discuss now even cycles, as well as certain families of paths.
	
	 
		Let C_k be any edge-ordered cycle graph on k ≥ 5 vertices. Then sat_m(n,C_k) = Θ(n).
	
	
	
		Let H be the host graph obtained from the construction described in Theorem <ref> and let V = {v_1, ..., v_n-k+2} be the independent set in it. Let u_1u_2 be the minimal edge in C_k, and let u_0 (resp. u_3) be the other neighbor of u_1 (resp. u_2). Assume there is a copy of C_k in H, then clearly there should be at least two vertices from V in such a cycle, v_i and v_j. Note that both of these vertices have degree two, so both of their incident edges have to be in the copy of C_k. However, they are both connected to the vertices u_0 and u_3. Therefore, we get that v_i, v_j, u_0, u_3 form a 4-cycle, but since we assumed that k ≥ 5, we get a contradiction.
	
	
	 
		Let P_k be a path on the vertex set {u_1, u_2, ...,u_k} for k ≥ 4, such that l(u_2u_3) ≠ 0 and l(u_k-2u_k-1) ≠ 0, where 0 is the minimal edge-label. Then sat_m(n,P_k) = Θ(n).
	
	
	
		Following the same procedure as in Theorem <ref>, we only need to show that the host graph is P_k-free. We will show that its underlying graph does not have any path on k vertices, regardless of its labels. 
		
		Consider first the case when k ≥ 6 and l(u_iu_i+1) = 0, for some i ∈{3,...,k-3}. Then after we delete the vertices u_i and u_i+1, we get two paths u_1u_2...u_i-1 and u_i+2u_i+3...u_k. Then we connect u_i-1 and u_i+2 with all the vertices in the set V = {v_1,..., v_n-k+2} and we obtain the host graph H. 
		
		Assume on the contrary, that there is a copy of P_k in H. Then there must be at least two vertices from V in such a copy. Moreover, these two vertices must be connected either through u_i-1 or u_i+2. Suppose wlog. it is u_i+2, but then none of the vertices u_j, j > i+2 can be in the copy of the path P_k. However, this means that our path can have at most i+2 vertices, which is not enough since we assumed that i ≤ k-3. Contradiction.
		
		It remains to check the case when l(u_1u_2) = 0 (or by symmetry l(u_k-1u_k) = 0). We can see that again two vertices from the independent set V are necessary, and they are connected through u_3. But this only gives a copy of P_3, while we assumed that k ≥ 4. So, we get a contradiction again.
	
	
	Using a very similar construction to the one in Claim <ref> that was used to prove the linearity of the saturation function of D_4, we are able to generalize this result in terms of the semisaturation function.
	
	 
		Let G be an edge-ordered graph with labels l: E(G) →ℕ and let ab be the minimal edge in G, such that for all w ∈ N_G(a) ∩ N_G(b) we have l(aw) = l(bw)+1. Then, ssat_m(n,G) = Θ(n).
	
	
	
		Assume that G has k+2 vertices for some k ∈ℕ. Then we create H starting with a copy of G ∖{u,v} keeping the same labels and n-k isolated vertices denoted by v_1, ...,v_n-k. Then we add an edge between v_i and w for all 1 ≤ i ≤ n-k and w ∈ N_G(a) ∪ N_G(b) and label them with 
		
    L(v_iw) =
    			
    				l(aw).i,    if  w ∈ N_G(a)
    				
    
    				l(bw).i,    if  w ∈ N_G(b) ∖ N_G(a).

		
		As usual, it is simple to check that H semisaturates G for the edges inside K_n-k and then we can greedily add at most a linear number of edges to get a semisaturating host graph.
	
	
	Claim <ref> served as a warm-up to the following more general statement, for which first we need to introduce some notations.
	
	
		Let l(E') = {l(e)  e ∈ E'}, where E' ⊆ E(G). We say that l(E') forms an interval in l(G) if there is no other edge e ∈ E(G) such that l(e_1) < l(e) < l(e_2), for some e_1, e_2 ∈ E'.
	

	We write l(E') < l(E”), if l(e') < l(e”) for all e' ∈ E' and e”∈ E”. In Remark <ref>, we discussed the labeling of graphs for which Theorem <ref> cannot be applied.
	In the following, we treat a special case of this family.
	
	
		Let G be an edge-ordered graph with labels l: E(G) →ℕ and e_0 = ab its minimal edge. We denote by {u_1, u_2, ..., u_k} := N_G(a) ∩ N_G(b) and we write l(au_i) = a_i and l(bu_i) = b_i for all 1 ≤ i ≤ k. We assume that a_1 < a_2 < ... < a_k and b_1 < b_2 < ... < b_k. Let 
    A_1 = {a_1, ..., a_c_1}, A_2 = {a_c_1+1, ..., a_c_2},..., A_d = {a_c_d-1+1, ..., a_c_d},

		
    B_1 = {b_1, ..., b_c_1}, B_2 = {b_c_1+1, ..., b_c_2},..., B_d = {b_c_d-1+1, ..., b_c_d},

		for some 0 = c_0 ≤ c_1 ≤ ... ≤ c_d = k. If  for all 1 ≤ i ≤ d A_i < B_i and A_i ∪ B_i form an interval in l(G), then ssat_m(n,G) = Θ(n). See Figure <ref> for an example.
	
	
		Assume that G has m+2 vertices for some m ∈ℕ. We create H from a copy of G ∖{a,b}, keeping the same labels and from n-m isolated vertices denoted by V = {v_1, ...,v_n-m}. Then we add an edge v_iw, for all 1 ≤ i ≤ n-m and w ∈ N_G(a) ∪ N_G(b) and label them as follows:
		
    L(v_iw) =
    			
    				a_c_s+1.i.j,    if  w = u_j, c_s+1 ≤ j ≤ c_s+1,
    				
    
    				l(aw).i,    if  w ∈ N_G(a) ∖ N_G(b),
    				
    
    				l(bw).i,    if  w ∈ N_G(b) ∖ N_G(a).

		Adding an edge in V creates a copy of G with some labeling. It remains to check that this copy is labeled as in G. We add an edge v_iv_j to H, for 1 ≤ i < i' ≤ n-m. The edge v_iv_i' has to play the role of ab in the copy of G. It is enough to check that the neighbors of v_i and v_i' are labeled correctly. If w ∈ N_G(a)  N_G(b), then l(aw).i is mapped to l(aw) (resp. l(bw).i is mapped to l(bw)). If w ∈ N_G(a) ∩ N_G(b), then for all 0 ≤ s ≤ d-1 the interval {a_c_s+1.i.(c_s+1), ..., a_c_s+1.i.c_s+1, a_c_s+1.i'.(c_s+1),..., a_c_s+1.i'.c_s+1} is mapped to the interval {a_c_s+1, ..., a_c_s+1,b_c_s+1, ..., b_c_s+1}. For example, the interval {a_2, a_3, b_2, b_3} in Figure <ref>, is given in the host graph by the interval {6.i.2, 6.i.3, 6.i'.2, 6.i'.3}.
		
		We conclude that H indeed semisaturates G for a non-edge in the independent set V.
		Finally, we can add linearly many edges to H to get a semisaturating host graph.
	
	
	

	Extremal functions of edge-ordered graphs. As we mentioned in Remark <ref>, the extremal function ex_e(n,G) provides an upper bound for every definition of saturation function. Kucheriya and Tardos <cit.> characterized connected edge-ordered graphs with linear extremal function. Note that any connected graph has at least linear saturation function by Theorem <ref>. This means that we can apply their results to get more examples of graphs with linear saturation function. In particular, their results include trees with certain edge-orderings, which we did not discuss here. By Corollary <ref> we get that any edge-ordered tree (which is not a single edge) has a linear semisaturation function. However, we do not have the same result for saturation. On the other hand, the extremal functions for paths already have more complicated behaviour. It was shown in <cit.> that extremal functions of certain paths on four edges can be Θ(nlog n), or even n2, while we just showed that for almost all paths the saturation function is linear. From Corollary <ref> we get that edge-ordered forests have linear semisaturation functions, we do not know if the analogous result holds for saturation functions. It was shown in <cit.> that for any edge-ordered star forest we have the almost linear upper bound ex(n,G)= n2^(α(n))^c, where α(n) is the inverse Ackermann function.
	
	

§ BOUNDED SATURATION FUNCTIONS

	
	In this section, we only deal with sat_m and ssat_m, the other variations are discussed in later sections. Claim <ref> characterizes edge-ordered graphs with bounded ssat_m, they are exactly those in which e_0 is isolated. It may actually be true that the same holds for sat_m. We consider special cases of edge-ordered graphs with isolated e_0 and prove that they have bounded sat_m.
	
	
	The saturation function of vertex-ordered matching graphs showed non-trivial behaviour and its characterization is still unknown <cit.>. However, for edge-ordered matchings it is rather simple to describe their behaviour, mainly due to the fact that for a given size there is only one matching up to isomorphism, which is not the case for vertex-ordered graphs. Therefore, the saturation number of any edge-ordered matching will be the same as the saturation number of its underlying graph. We can make a similar observation for star graphs, as up to isomorphism we can label its edges in only one way, so its saturation number is equal to the one of its underlying graphs. Saturation numbers of matchings and stars were exactly determined by Kászonyi and Tuza <cit.>. We recall these results below.
	
	 <cit.> 
		Let k,t ≥ 1 and n large enough. Then
		
    sat(n,tK_2) = 3t-3  and  sat(n,K_1,k) = k-1/2n- 1/2⌊k^2/4⌋.

	
	
	So, the matching graphs provide an infinite family of edge-ordered graphs with bounded saturation number. However, by introducing the notion of monotone disjoint union of graphs, we will be able to find more general families with bounded saturation number.
	
	VB: It is quite likely that the results about supporter graphs are redundant, as they are already included in the result about monotone forests
	
	
		We say that G is a monotone disjoint union of graphs, if for any two connected components C', C” of G either l(e') < l(e”) or l(e') > l(e”) for all e' ∈ E(C') and e”∈ E(C”).
	
	
	
		We say that P_k is a monotone path, if its underlying graph is a path on k vertices v_1, ..., v_k and its edges are labeled by l(v_iv_i+1) = a_i such that a_1 < ... < a_k-1.
	
	
	
		Let G be a connected edge-ordered graph on t vertices. We say that S(G) is a supporter graph of G if it has t+1 vertices and S(G)-v has a subgraph isomorphic to G for every v ∈ V(S(G)).
	
	
	
	
		Let F = e_0 + G, where e_0 denotes the isolated minimal edge and G=⋃_i=1^k G_i is a monotone disjoint union of connected graphs such that for each 1 ≤ i ≤ k, there exists a supporter graph S(G_i). Then sat_m(n,G) = O(1).
	
	
	
		Assume that F has r+1 vertices. We take a monotone disjoint union ⋃_i=1^k S(G_i) + K̅_n-r. Then, we take a vertex in each S(G_i) and we merge these k vertices into one to obtain a host graph H on n vertices. The graph H has only r non-isolated vertices, therefore it is F-free. Moreover, by taking any vertex u ∈ V(K̅_n-r) and connecting it with any other vertex v ∈ V(H), we obtain a copy of F, where uv plays the role of e_0. Thus, H is also F-saturated and it has a bounded number of edges.
	
	
	
	
	
	A natural problem motivated by Proposition <ref> is to classify which graphs have a supporter graph or at least find large classes of graphs that have a supporter graph. The next lemma shows three such classes.
	
	
		If G is any edge-ordered star, triangle or a monotone path, then S(G) exists.
	
	
	
		As we mentioned, any two edge-ordered stars on r vertices are isomorphic. Therefore, we can take for a supporter graph K_r+1 with any edge-ordering. If we remove any vertex from it, the remaining graph will contain a star on r vertices.
		
		Similarly for a triangle, we take K_4 with any edge-ordering. Deleting any vertex from it, we get a triangle.
		
		Finally, for a monotone path P_k-1 on k-1 vertices, in this case it is less obvious how to make S(P_k-1). We start by taking a cycle C_k, and we label its edges as follows: l(v_iv_i+1)=2i-1, for all 1 ≤ i ≤ k (where we assume v_k+1=v_1). Then we add to it the edges v_iv_i+2 that have labels l(v_iv_i+2) = 2i-2 for 2 ≤ i ≤ k+1 (we assume v_k+1=v_1, v_k+2=v_2 and v_k+3=v_3). See Figure <ref> for an example of a supporter graph when k=8. It is not hard to check that this is actually a supporter graph of P_k-1. First, assume that we have removed a vertex v_j such that j ∈{1, 2, k}. Then, the path v_j+1v_j+2...v_j+k-1 is a monotone path on k-1 vertices. On the other side, if j ∈{ 3, 4, ..., k-1}, then the path v_1v_2...v_j-1v_j+1...v_k is a desired monotone path on k-1 vertices.
	
	
	
	
	 
		Let F = e_0 + G, where G is a monotone disjoint union of stars, triangles and monotone paths. Then sat_m(n,F) = O(1).
	
	
	
	
	As we showed in Theorem <ref>, the saturation function can be bounded only if the minimal edge of a graph G is isolated. In the following, we consider a construction which will help us prove that for many edge-ordered graphs the saturation function is indeed bounded.
	
	
		Let G be an edge-ordered graph whose edges are labeled with the set [m]. Take three copies of G, which are denoted by G_1, G_2 and G_3, such that each G_i is labeled with a map k → 3(k-1)+i, for k ∈ [m] and i ∈{1,2,3}. We denote by a_ib_i the corresponding minimal edge of G_i. Finally, we obtain T(G) by merging the following pairs of vertices: a_1 and b_3, a_2 and b_1, a_3 and b_2.
	
	
	
		Let G be an edge-ordered graph labeled with the set [m], and let H be the union of T(G) and a set of isolated vertices. Then H semisaturates e_0+G for any non-edge uv, where at least one of the vertices is isolated.
	
	
		Assume that u is an isolated vertex and v is arbitrary. Then there must exist i ∈{1,2,3} such that v ∉ V(G_i). Therefore, uv + G_i contains a copy of e_0 + G.
	
	
	Therefore, to show that sat_m(n,e_0+G) is bounded for some G, it is enough to verify that T(G) does not contain e_0 + G.
	
	
		We say that G is an A-B graph if it is obtained from two disjoint connected graphs G_A and G_B and adding a minimal edge e_1 = ab such that a ∈ V(G_A) and b ∈ V(G_B).
	
	
	
	
	 
		Let F = e_0 + G, where G is an A-B graph labeled with the set [m]. Then sat_m(n,F) = O(1).
	
	
	
		We consider a host graph H which is a union of T(G) and isolated vertices. We also assume that a_ib_i connects the components G_i,A and G_i,B, as shown in Figure <ref>. 
		
		To show that H is F-free, we assume the contrary. First, we consider the case when none of the three minimal edges of H (that is, the edges a_1a_2, a_2a_3, a_3a_1) are contained in the copy of F. Notice that by deleting these edges from H we would get a graph with three connected components, such that each of them has |V(G)| - 1 vertices. Therefore, H cannot contain F in this case. Otherwise one such edge, wlog. a_1a_2 is contained in the copy of F and has to play the role of the minimal edge e_0 in the copy of F. However,  H ∖{a_1, a_2} has connected components only of size at most |V(G)|-1 and thus it cannot contain G, which gives a contradiction.
	
	
	We can easily see that any edge-ordered tree is an A-B graph.
	
	 
		Let T be an edge-ordered tree, then sat_m(n, e_0 + T) = O(1).
	
	
	
		Let H(F) be a saturation graph of F such that sat_m(n,e_0 + F) = O(1). Let G be an edge-ordered graph, such that l_min(G) < l_min(F). [Here l_min(G) denotes the minimal label of G.].Then H(F) avoids G+F.
	

	Using this observation, we can extend Theorem <ref> to certain disconnected graphs.
	
	
		Let G = ∑_i=1^k G_i be a disjoint union of A-B graphs such that, |V(G_i)|≤|V(G_i+1)| and l_min(G_i) < l_min(G_i+1) for all 1 ≤ i ≤ k-1. Then it follows that sat_m(n,e_0 + G) = O(1).
	
	
	
		Let H be a union of a set of isolated vertices and of ∑_i=1^k T(G_i) such that for edges e_i∈ T(G_i) and e_j∈ T(G_j) we have l(e_i)<l(e_j). It is simple to check that H indeed semisaturates e_0+G apart from O(1) edges. To show that it also avoids e_0+G, we assume the contrary and proceed by induction. The case k=1 follows by Theorem <ref>. Then we assume that the claim holds for all 1 ≤ j ≤ k-1. 
		
		We distinguish two cases. First, assume there is a copy e_0+G in H such that G_k is in T(G_k). In that case, by the inductional hypothesis either e_0 or  another G_j, for j<k is also in T(G_k), but this is not possible by Observation <ref>. Second, assume there is a copy of G_k in T(G_j) for some j<k. Since we assumed that |V(G_j)|≤|V(G_k)|, it follows that the copy of G_k contains at least one of the edges in T(G_j) that correspond to the minimal edge of G_j, since by deleting these three edges we get three connected components of size |V(G_j)|-1. This implies that ∑_i=1^j-1 T(G_i) contains a copy of e_0 + ∑_i=1^j-1 G_i, which is not possible by the inductive hypothesis.
	
	
	We identify several other families of edge-ordered graphs for which T(G) allows us to show that their saturation functions are bounded.
	
	
		Let F = e_0 + G, such that G is connected and there exist no pair of vertices u,v ∈ V(G) for which at least one of the graphs G ∖{u,v}, G ∖{u}, G ∖{v} is disconnected. Then sat_m(n,F) = O(1).
	
	
		We use again the same construction T(G) as a host graph and we need to show that it avoids F. Notice first that a copy of G cannot be identically any of the graphs G_1, G_2 and G_3. So, it means that the copy of G has to be found in the union of either two or three of these graphs. Also a copy of G cannot contain any of the three edges a_1a_2, a_2a_3, a_3a_1, as there is no edge in that case which has the role of e_0.
		
		Assume wlog. that G is in the union of G_1 and G_2, then a_3 is certainly in the copy of G, otherwise the graph would not be connected. But then a_3 is a vertex separator, which is not possible by assumption. We then assume that G can be contained in all three copies. In that case all three vertices a_1, a_2, a_3 have to be in the copy of G. Moreover, since G is connected there should be at least two vertices among them, wlog. a_1 and a_2 such that there is a path of length at least two between them (recall that the edge a_1a_2 is not in the copy of G). So, if we delete the vertices a_1 and a_2, the copy of G becomes disconnected which contradicts our assumption.
	
	
	 
		Let K_r be a complete graph for r ≥ 2 with arbitrary labeling. Then sat_m(n,e_0 + K_r) = O(1).
	
	
	
		Let F = e_0 + G, where G is a connected graph whose minimal edge is ab and deg(a) ≥ deg(b) > deg(v) for all v ∈ V(G) ∖{a,b}. Then sat_m(n,F) = O(1).
	
	
		We again want to show that T(G) is F-free. Notice that the only vertices that have large enough degrees to have the role of the vertices a and b, are the vertices a_1,a_2 and a_3. So, the minimal edge of G must connect two of these vertices, but then there is no edge in T(G) that can play the role of e_0, which leads to a contradiction.
	
	
	We now identify another family with bounded saturation function, but this time without using the construction T(G). Instead of three copies of the graph G, now we use only two and add a few additional edges.
	
	 
		Let F = e_0 + G, such that there exists a vertex v ∈ V(G), which we call the peak, of degree d ∉{0,2} whose incident edges are labeled with the set {1,...,d} (the rest of the edges get larger labels) and G-v is connected. Then sat_m(n,F) = O(1).
	
	
	
		We proceed by constructing a host graph H that semisaturates F. We first join two copies of G at the peak v and we denote the two disjoint parts by G' and G”. Then we consider the neighbors {v_1, v_2, ...,v_d}⊂ V(G') of v and we label each edge l(vv_i) = 3(i-1)+1 for 1 ≤ i ≤ d. Additionally, we label by l(vu_i) = 3(i-1)+2, where u_i is the neighbor of v in G” for all 1 ≤ i ≤ d. Moreover, we add the edges between u_1 and all the v_i and we label them with l(u_1v_i) = 3i. The remaining edges in G' and G” can be any labels greater than 3d that respect the edge-order inherited from G. See Figure <ref> for a construction. As usual, the remaining vertices are isolated.
		
		
		
		First, we want to show that H is F-free. We treat the case d=1 separately by observing that no matter which edge we take to play the role of e_0, the copy of G must be contained in either G' or G”. That is not possible, since G is larger than any of them. From now on, we consider the case d ≥ 3. Assume on the contrary that H is not F-free and assume that in a copy of F we have a peak p ∉ N_H[v]. In that case, we can notice that none of the edges connecting two vertices of N_H[v] can be in the copy of G. However the rest of the graph H contains two connected components with less than |V(G)| vertices, a contradiction. 
		
		Let p=v be the peak, in that case one of the edges u_1v_1,...,u_1v_d must play the role of e_0 in the copy of F. Thus, if we remove v from the copy of G, we get a disconnected graph, as some of its neighbors are in G' and the other are in G” contrary to our assumption that G-v is connected. 	
		
		Assume now that the peak is p=v_i for some 1 ≤ i ≤ d. As G is larger than G', the copy of G has a vertex outside G'. The peak p must have at least one neighbor in G' (as it has only two further incident edges in H) and such an edge has larger label than any edge  between G', p, and G”. Therefore, no edge in the copy of G-v can use the edges between G', p, and G” and so G-v is a disconnected graph, a contradiction. A similar argument works for p ∈{u_2, ..., u_d}. 	
		It remains to check if u_1 can be the peak. Clearly, u_1v and u_1v_1 cannot be in the copy of G as there would be no choice for the minimal edge. Thus, one neighbor of the peak has to be in G”, and for the same reason as in the previous case, we get that u_1 cannot be the peak.
		
		It remains to show that H is F-saturated for the non-edges incident to isolated vertices. Indeed, if we take an isolated vertex and connect it with any vertex in G' or G”, we get a new copy of F. If we connect it with v, then u_1 can have the role of the peak and together with G' we get a copy of F.
		
		Finally, we can add a bounded number of edges greedily between non-isolated vertices to get a required host graph.		
	
	
	
		Let F = ∑_i=1^k F_i be such that l(F_i) < l(F_j) for all 1 ≤ i<j ≤ k and sat_m(n,e_0 + F_i) = O(1), for all 1 ≤ i ≤ k, where F_i are connected. Then it follows that sat_m(n,e_0 + F) = O(1).
	
	
	
		Let H be a union of ∑_i=1^k H(F_i) and isolated vertices, where H(F_i) is the non-isolated part of the saturation graph of F_i. Additionally we label H such that l(H(F_i)) < l(H(F_j)) for all 1 ≤ i<j ≤ k. It is simple to see that H semisaturates e_0 + F apart from O(1) edges. So, it remains to check that it also avoids e_0 + F. 
		
		We proceed by induction on k. The case k = 1 follows directly by assumption. Assume on the contrary that for k the claim fails.
		As the claim holds for k-1 by inductive hypothesis, ∑_i=1^k-1 H(F_i) contains no copy of e_0 + ∑_i=1^k-1 F_i. This implies that there is a copy of F_j in H(F_k) for some j ≤ k-1, and clearly a copy of F_k in H(F_k) since l(F_j) < l(F_k). But this is not possible by Observation <ref>.
	
	
	The previous result allows us to take a monotone union of any connected graphs that we previously showed to have bounded saturation function. Among others, this generalizes Corollary <ref>.
	
	
		Let F be a monotone forest if F = ∑_i=1^k T_i such that every T_i is a tree and l(T_i) < l(T_j) for all 1 ≤ i<j ≤ k. 
	
	
	 
		For every monotone forest F, we have sat_m(n,e_0 + F) = O(1).
	

	

§ SAT_S AND SSAT_S
 
	
	Recall that in Definition <ref>, we stated two more definitions of saturation function. The reason why we do not have a unique definition is that we have a choice on how to label the missing edge. So far, we worked with the definition where a new edge always gets the minimal label.
	
	Another interesting definition of saturation allows a new edge to introduce a copy of a given edge-ordered graph G for some label and we denoted this saturation function by sat_s(n,G). As we already mentioned at the beginning sat_s(n,G) ≤ sat_m(n,G), which means that all the upper bounds that we proved for sat_m definition hold for sat_s as well. This also includes the general O(n log n) upper bound for semisaturation. Again, with an analogous proof it follows that there is no dichotomy in this framework either. However, this time the result is obtained for a more restricted family of graphs:
	
	
		Let G be an edge-ordered graph. If for every edge ab ∈ E(G) there exist v,w ∈ N_G(a) ∩ N_G(b) such that l(av) < l(aw) and l(bv) > l(bw), then sat_s(n,G) ≥ ssat_s(n,G)=Ω(n√(log n)). 
	
	
	Notice that none of the diamond graphs shown on Figure <ref> satisfy this property. However, we can take a K_4 and label it as shown on Figure <ref>. Moreover, for K_k with k≥ 5 the following edge-order is also suitable: label the vertices with [k] and then for every i j in [k] the edge ij gets label min(|j-i|, k-|j-i|), that is, the cyclic distance between i and j when taken modulo k. Then we perturb the labels slightly so there are no equal labels on the edges. It is easy to check that this edge-order satisfies the assumption of Corollary <ref> for every k≥ 5.
	
	
		For every k≥ 4, there exists an edge-ordered graph G_k with underlying graph K_k, such that sat_s(n,G_k) ≥ ssat_s(n,G_k)=Ω(n√(log n)).
	
	
	
	
	Note that, from the inequality sat_s(n,G) ≤ sat_m(n,G) we get that whenever a graph G has linear sat_m function and it contains no isolated edges, then it also has linear sat_s function. This implies that the cycles and paths we discussed in Claims <ref> and <ref> also have linear saturation function for this definition. Moreover, we can use the same approach as in Theorem <ref> to find an even larger family of edge-ordered graphs with linear semisaturation function.
	
	 
		Let G be an edge-ordered graph which has no isolated edges. If there exists an edge ab ∈ E(G) such that N_G(a) ∩ N_G(b) = ∅, then ssat_s(n,G) = Θ(n).
	
	
		
	

§ SAT_E AND SSAT_E

	
	The most natural saturation definition for edge-ordered graphs seems to be sat_e(n,G) where we allow the new edge to have any label inserted in the linear order of the edges of the host graph H. In this section we state several results about this variant. We left these to the be last as most of the forthcoming results follow from the results that we have already proved.


	Recall that for all edge-ordered graphs we have sat_m(n,G) ≤ sat_e(n,G). Therefore, all the lower bounds that we showed for sat_m also hold for sat_e definition. Most importantly, this gives us a family of edge-ordered graphs with superlinear semisaturation (and saturation) function. The next result follows directly from Theorem <ref>.
	
	 
		Let G be an edge-ordered graph. If ab ∈{e_0, e_max} and there exist two vertices v,w ∈ N_G(a) ∩ N_G(b) such that l(av) < l(aw) and l(bv) > l(bw), then sat_e(n,G) ≥ ssat_e(n,G)=Ω(n√(log n)).
	

	We now show the upper bound for the diamond graph D_0.

	 
		
    ssat_e(n,D_0) ≤ sat_e(n,D_0)=O(n log n/loglog n).

	

	
		We start by H = K_k,n, the host graph that we constructed in the proof of Theorem <ref>. Recall that H was labeled by ℝ_+. We add now two more vertices u_1 and u_2, and we connect them with the independent set in H that we denote by V = {v_1, …, v_n}. We label the edges as follows:
		
    l(u_1v_i) = -3.i  and  l(u_2v_i) = -1.i  for all  1 ≤ i ≤ n.
 We also add the edge u_1u_2 and label it by l(u_1u_2) = -2. Notice that by adding any edge between two vertices in V and labeling it by any positive number, we obtain a copy of D_0, where the new edge plays the role of bv, the maximal edge in D_0 (See Figure <ref>). Recall that from Theorem <ref>, if any edge in V is added with negative label or zero, a new copy of D_0 is created. Therefore we get a host graph that semisaturates D_0 for any non-edge in V for any real label.
		
		It remains to check that the host graph avoids D_0, so we assume the contrary. Notice that by deleting the edge u_1u_2, we get a complete bipartite graph. Therefore, since D_0 is not bipartite, u_1u_2 has to be in the copy of D_0. Even more, u_1u_2 has to play the role of the minimal edge ab. But that is not possible, because l(u_1v_i) < l(u_1u_2) for all 1 ≤ i ≤ n. This implies that u_1u_2 cannot be the minimal edge, which gives a contradiction. Finally, we can add greedily edges between k+2 vertices without introducing a copy of D_0, which does not affect the order of magnitude of the saturation function.
	
	
	Recall that by Theorem <ref> if G is an edge-ordered graph in which e_0 and e_max are isolated, then either sat_e(n,G) = O(1) or sat_e(n,G) = Ω(n). Otherwise, sat_e(n,G) = Ω(n). Here we show why Ω(n) was necessary in the statement already for the e_0+G+e_max part, a behaviour that is different from that of sat_m function in Theorem <ref>.
	
	 
		For every k≥ 5, there exists an edge-ordered graph G_k with underlying graph K_k, such that sat_e(n,e_0+G_k+e_max)=Ω (n√(log n)).
	
	
	
		Let G_k be an edge-ordered graph with underlying graph K_k. We denote F_k=e_0+G_k+e_max.
		Let H be a host graph saturating F_k for sat_e. Take a non-edge. Adding it as a new edge e with any label should introduce a copy of F_k. Note that there are only finitely many essentially different labels (one between each pair of consecutive labels in F_k), so we only concentrate on them. 
		
		We claim that there must be a label for which in the introduced copy of F_k the new edge e does not play the role of e_0 nor e_max of F_k. Assume on the contrary and take the smallest label such that e plays the role of e_max in a copy F'_k of F_k (this exists, as when we choose the largest label for e then e must play the role of e_max). Take the next smallest possible label for e (this again must exist as the current label could not have been the smallest), then e must play the role of e_0 in a copy F”_k of F_k. Let G'_k and G”_k be the K_k part of F'_k and F”_k, respectively. Notice that all the labels of the edges of G'_k are smaller than the labels of the edges of G”_k, except possibly of the edge that appears in both of them. Therefore, their vertex sets intersect in at most two vertices. In addition the edge that plays the role of e_max in F”_k can again have at most one common vertex with G'_k. Therefore, there are at least two vertices in G'_k, which are not in F”_k. This edge can also play the role of e_0 in F”_k instead of e, contradicting that H avoids F_k.
		
		Thus, for each non-edge we can add it with an appropriate label such that we get a copy of F_k in which the new edge e plays the role of an edge in G_k. That is, adding e with this label must introduce a new copy of G_k. This means that H is semisaturating G_k for ssat_s, therefore sat_e(n,e_0+G_k+e_max)=|H|≥ ssat_s(n,G_k). Applying Corollary <ref> gives that there is an appropriate G_k such that ssat_s(n,G_k) = Ω(n√(log n)), finishing the proof.		
	
	
	
	
	One might wonder if the main observation in the previous proof holds in some generality, that is, if sat_e(n,e_0+G+e_max)=Ω(ssat_s(n,G)) always. We have seen that this holds if G is an edge-ordered complete graph on at least 5 vertices. Here we give an example that shows that this is not true already for a graph G with two edges. Let G be the unique edge-ordered cherry graph (has two edges on three vertices). By Corollary <ref> ssat_s(n,G)=Ω (n). On the other hand we claim that F=e_0+G+e_max has sat_e(n,F)=O(1). Let H be the graph on n vertices such that n-8 of the vertices are isolated, on the remaining 8 vertices we have two K_4's, labeled as shown on Figure <ref>. It is easy to check that H avoids F yet adding any new edge incident to at least one isolated vertex with any label introduces a copy of F. Therefore, by greedily adding further edges between the two K_4's if necessary, we get a saturating host graph on n vertices and O(1) edges.
	
	Furthermore, notice that if two copies G' and G” of a graph G, such that l(G') < l(G”), can be edge-disjointly added on |V(G)|+1 vertices, then sat_e(n,e_0+G+e_max) = O(n). Indeed, the host graph can be obtained by adding a vertex disjoint cherry with minimal and maximal edge to this graph on |V(G)|+1 vertices (the two copies of G have labels that form two intervals of l(G)), as well as arbitrary many isolated vertices. This graph clearly avoids e_0+G+e_max and by adding an edge with any label between two isolated vertices we get a copy of e_0+G+e_max. Thus by adding O(n) edges we get a host graph. With this construction, we can produce many examples that have at most linear saturation function.
	
	We can observe that the analogue of Corollary <ref> also holds for sat_e definition. However, we need to add a condition that G - e_max also becomes bipartite. By gluing two host graphs together, we can see that we still keep the bipartite property, which means that the host graph remains G-free. See Figure <ref> for an example.

	 
		Let G be a non-bipartite edge-ordered graph. Let e_0 = ab and e_max = cd such that G-e_0 and G-e_max are bipartite and N_G(a) ∩ N_G(b) = N_G(c) ∩ N_G(d) = ∅, then sat_e(n,G) = Θ(n).
	

	
	
	From this we get the following explicit infinite family of graphs that have linear ssat_e and sat_e function. 

	 
		Let C_2k+1 be an edge-ordered cycle graph. Then for every k ≥ 1, 
    sat_e(n,C_2k+1) = Θ(n)  and  ssat_e(n,C_2k+1) = Θ(n).

	

	To finish this section, we also present an infinite family of edge-ordered graphs with bounded sat_e function. Since all edge-ordered matching graphs with fixed number of edges are isomorphic to each other, in this case saturation problem is equivalent to saturation problem of unordered graphs. Therefore, we get from <cit.> that:
	
	 
		Let M_k be an edge-ordered matching graph. Then 
    sat_e(n, M_k) = O(1)  , for every k ≥ 1.

	
	
	

§ OPEN QUESTIONS

	
	We conclude by highlighting the most interesting problems left open.
	
	We do not have any example of an edge-ordered graph with linear sat_m saturation function whose minimal edge is isolated. The construction T(G) (as in Figure <ref>) seems like a good candidate to show that such a graph does not exist. 
	
	
		Let F = e_0 + G, then sat_m(n,F) = O(1).
	
	
	Note that the analogous statement for sat_e definition, where we replace e_0 with e_0+e_max, is not true, as we have seen earlier.
	
	Theorem <ref> shows that if the endvertices of the non-isolated minimal edge of G have disjoint neighborhoods, then the semisaturation function of G is linear. So far, we do not have a counterexample and it is possible that this family has linear saturation function, as well.
	
		Let G be an edge-ordered graph and let ab be the minimal edge in G such that deg(a) ≥ 2. If N_G(a) ∩ N_G(b) = ∅, then sat_m(n,G) = Θ(n).
	

	The previous two conjectures raise a question whether in general for a given graph, the order of magnitude of sat_m and ssat_m are always the same. We do not have a counterexample for this claim. However, they do not have necessarily the same exact values, for example one can show that sat_m(n, M_k) = 3(k-1) and ssat_m(n, M_k) = k+1, for k ≥ 2. 
	For sat_e definition, we already have some counterexamples. Theorem <ref> and Theorem <ref> give edge-ordered graphs with bounded ssat_e function, but superlinear sat_e function.
	
	The main open question that remains, is to find a general upper bound for saturation function. We expect to have a non-trivial upper bound and we state it in terms of sat_e which implies the same  upper bound for sat_m.
	
	
		For every edge-ordered graph G, we have sat_e(n,G) = O(n^1+o(1)).
	

	It seems plausible to expect that the same upper bound as for semisaturation can be proven. Therefore, we also state the following stronger conjecture.
	
	
		For every edge-ordered graph G, we have sat_e(n,G) = O(n log n).
	

	Besides that, it might be possible that the general upper bound depends on the chosen definition. 
	
	We can check that the construction given in the proof of Theorem <ref>, indeed gives a saturation upper bound for many graphs. However, for some cases, such as certain families of trees, the construction only gives a semisaturation upper bound. 

	The semisaturation problem can be studied separately and we managed to understand the behaviour of semisaturation functions slightly better. The major question is whether the upper bound can be improved, or at least can we find an example when the bound is achieved.
	
	
		Is there an edge-ordered graph G such that ssat_e(n,G) = Θ(n log n)?
	

	Studying thoroughly the diamond graph D_0 was helpful to understand how the saturation functions behave for more general families of edge-ordered graphs. In fact, we were able to extend the lower and upper bounds to infinite families of graphs. One possible direction is to study further the saturation function of D_0, which would again most likely lead to a more general result. On the other hand, D_0 is the only connected edge-ordered graph for which we showed an upper bound O(n log n/loglog n). Can we extend this result to a more general family of connected edge-ordered graphs?
	
	We were able to show at least three different behaviours of saturation functions. A natural question is if there are more than that, or more generally, are there finitely or infinitely many different orders of magnitude for saturation functions of edge-ordered graphs? Analogous question for semisaturation functions is also open.
	
	Recently, the saturation functions of sequences were studied <cit.>. They showed that the saturation function is either bounded or at least linear. It would be interesting to investigate whether an example with a superlinear saturation function can be found, or to show that the usual dichotomy holds. Regarding semisaturation functions, they showed that they are at most linear, which is not the case for edge-ordered graphs.
	
	Acknowledgements
	
	Part of this research was performed while the first author was visiting the Institute for Pure and Applied Mathematics (IPAM), which is supported by the National Science Foundation (Grant No. DMS-1925919).
	
	Research supported by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences, by the National Research, Development and Innovation Office – NKFIH under the grant K 132696 and FK 132060, by the ÚNKP-23-5 New National Excellence Program of the Ministry for Innovation and Technology from the source of the National Research, Development and Innovation Fund and by the ERC Advanced Grant “ERMiD”. This research has been implemented with the support provided by the Ministry of Innovation and Technology of Hungary from the National Research, Development and Innovation Fund, financed under the  ELTE TKP 2021-NKTA-62 funding scheme.
	
	plainurl
	
	


 Paper Title: On set systems without singleton intersections.



A HD thermal irradiated wind from outer thin accretion disk in LLAGNs
    Nagendra Kumar 12nagendra.bhu@gmail.com; * current address
    
=====================================================================



§ ABSTRACT 
Consider a family ℱ of k-subsets of an ambient (k^2-k+1)-set such that no pair of k-subsets in ℱ intersects in exactly one element.   
In this short note we show that the maximal size of such ℱ is k^2-k-1k-2 for every k > 1. 


Keywords: Johnson scheme, Erdős–Sós problem, finite projective planes.



§ INTRODUCTION



A large branch of combinatorics grows from the celebrated Erdős–Ko–Rado theorem <cit.>, which states that
a family of pairwise intersecting k-element subsets of an n-element set has size at most n-1k-1 provided that n ≥ 2k. 

For our aim it is convenient to define a Johnson graph J(n,k,t), whose vertices are k-element subsets of an n-element set and edges connect pairs of vertices with intersection t. An independent set is a vertex subset of a graph such that it does not contain an edge.
Let α(G) stand for the size of a maximal independent set in a graph G.


In this language the statement of the Erdős–Ko–Rado theorem is 

    α (J[n,k,0]) = n-1k-1

for n ≥ 2k. The problem of finding α (J[n,k,t]) is known as Erdős–Sós forbidden intersection problem.
The bibliography on this problem is wide and the proofs use very different techniques. Let us briefly provide the highlights.
Frankl and Füredi <cit.> used so-called Δ-system method to show that 
α(J[n,k,t]) = n-tk-t for n > n_0(k) and k ≥ 2t+2.

Another very general result was obtained by Frankl and Wilson <cit.> by a rank bound: it gives 
α(J[n,k,t]) ≤nk-t-1 for k > 2t and k-t being a prime power.
This result has important applications to discrete geometry, see <cit.>. 

Recently, Ellis, Keller and Lifshitz <cit.> used junta-method to determine α(J[n,k,t]) for 
ε < k/n < 1/2 - ε and n > n_0(t,ε). 
Kupavskii and Zakharov showed that α(J[n,k,t]) = n-t-1k-t-1 for k > k_0, n =  k^α, t =  k^β, where α > 1 and 1/2 > β > 0 satisfy α > 1 + 2β. They use spread approximation technique <cit.>.


Also let us mention that α(J[n,4,1]) = n-22 for n ≥ 9, see Keevash–Mubayi–Wilson <cit.>.


Our contribution is the following

 
    For every k > 1 one has
    
    α(J[k^2-k+1,k,1]) = k^2-k-1k-2.




Note that for k > k_0 Theorem <ref> follows from the mentioned result of Kupavskii and Zakharov <cit.>.






§ TOOLS





 §.§ Johnson scheme and Bose–Mesner algebra


The facts from this subsection can be found in books <cit.> and <cit.>.


An associative scheme is a pair (V,ℛ) consisting of a finite set V and a family of non-empty binary relations ℛ on V, and satisfying the following properties:

     
  * sets R ∈ℛ form a partition V^2;
     
  * the diagonal Δ(V) of the set V^2 is an element of ℛ;
     
  * the set ℛ is closed under the interchange of the first and second coordinates in V^2;
     
  * for arbitrary relations R, S, T ∈ℛ numbers
     
    | v ∈ V : (u,v) ∈ R, (v,w) ∈ S|

     are the same for all (u,w) ∈ T.


The classical Johnson scheme is given by 

    V = [n]k;    R_i := {(v,u) ∈ V× V : ⟨ v,u ⟩ = k-i},     i=0,… k.




The Bose–Mesner algebra of the Johnson scheme is the algebra of nk×nk matrices, with entries defined by A(x,y) := f(|x ∩ y|). Since this is indeed algebra, these matrices are simultaneously diagonalizable.
A standard basis of the Bose–Mesner algebra is formed by matrices B_i(x,y) := |x∖ y|i, i = 0,…, k.
The eigenvalues of B_i are given by

    μ_j^(i) = (-1)^j k-ji-jn-i-jk-j.

Then the machinery offers to represent any matrix A from the Bose–Mesner algebra as ∑ b_iB_i and get it spectrum as 

    λ_j = ∑_i=0^k b_i μ_j^(i).



Let I be a subset of [n]k and χ_I stand for its characteristic vector. Denote by c_i, i=0,…, k the coefficients in the decomposition of χ_I with respect to the common eigenspaces of the Bose–Mesner algebra. 




 §.§ Hoffman bound


The following celebrated theorem is widely known as Hoffman ratio bound <cit.>.


    Let A be a pseudoadjacency matrix of a d-regular N-vertex graph G with non-negative entries. Then 

    α(G) ≤ N -λ_min/d - λ_min,

where α(G) is the independence number of G.


The proof is a one-line collection of several simple observations. Let I be any independent set in G and χ_I stand for its characteristic vector. Then

    0 = (Aχ_I,χ_I) = ∑_i=1^N a_i^2 λ_i ≥ a_1^2 d + (a_2^2 + … + a_N^2) λ_min = |I|^2/N d +  (|I| - |I|^2/N) λ_min,

where a_i are the coefficients in the decomposition of χ_I in the eigenbasis of A.
Here we use that a spectral radius of a d-regular graph is d and it is achieved at the all-unit vector.
Also in the case of an edge-transitive graph Hoffman bound coincides with Lovász theta-bound <cit.>.





§ THE PROOF



  An example of an independent set of size k^2-k-1k-2 is given by a collection of all sets, containing elements 1 and 2.

Let A be the adjacency matrix of the Johnson graph J(n,k,1).  Clearly, this matrix is given by f(1) = 1, f(0) = f(2) = … = f(k) = 0. It is straightforward to check that the coefficients in the standard basis of the Bose–Mesner algebra are the following
    
    b_0 = b_1 = … = b_k-2=0,    b_k-1 = 1,    b_k = -k

    and
    
    λ_0 = -kn-kk + kn-k+1k = k n-kk-1,

    
    λ_1 = kn-k-1k-1 - (k-1)n-kk-1,   λ_2 = -kn-k-2k-2 + (k-2)n-k-1k-2.

For n = k^2-k+1 we have

    λ_1 = λ_2 = -1/k-1n-kk-1 < 0

and

    λ_3 = kn-k-3k-3 - (k-3)n-k-2k-3 = 2k^2-3k-3/k^2-3k+2n-k-3k-3 > 0.

Also  |λ_4|, |λ_5|, …, |λ_k| ≤ k n-kk-3, hence λ_1 = λ_2 are the smallest eigenvalues.

Now the upper bound follows from the Hoffman bound (<ref>):

    -λ_min/d - λ_min = 1/k^2-k+1 = n-2k-2/nk.

    





§ DISCUSSION


Let us briefly discuss the sporadic nature of the result. In all solved cases maximal independent sets form designs or juntas.
Hoffman bound is tight when the corresponding characteristic vector belongs to maximal and minimal eigenspaces and it seems difficult to modify the method in other cases.
The maximal eigenspace is always unique and corresponds to the all-unit vector.
For t ≥ 1 the characteristic vectors of all known examples belong to more than two eigenspaces, so several minimal eigenvalues should coincide in order to use Hoffman bound. Summing up, it seems that the only case is t=1, in which we have an example with the characteristic vector in the first three eigenspaces.
So we need λ_1 = λ_2 which implies n = k^2 - k + 1.





 §.§ Finite projective planes


If k-1 is a prime power, then one can prove the upper bound in Theorem <ref> combinatorially. Since Johnson graphs are vertex-transitive (moreover they are edge-transitive), one has
    
    w(J) ·α(J) ≤ |V(J)|.

    In our case k^2 - k + 1 is the size of a projective plane over GF(k-1), and so w(J[k^2-k+1]) ≥ k^2-k+1.
    This immediately implies the bound.

    However for a composite k-1 the corresponding construction may not exist.
    A major negative result is a celebrated Bruck–Ryser theorem <cit.> which states that if n is a positive integer of the form 4k + 1 or 4k + 2 and n is not equal to the sum of two integer squares, then n does not occur as the order of a finite plane.  A widely known conjecture is that the order of a finite plane is always a prime power. Also, the non-existence of a finite plane of order 10 was proven by Lam <cit.>.





 §.§ Uniqueness


For k = 3 the graph J(7,3,1) has a maximal independent set of another structure, namely 

    {{1,2,3},{1,2,4},{1,3,4},{2,3,4},{5,6,7}}.

For other values of k it seems very likely that every maximal independent set of J(k^2-k+1,k,1) forms a family with two elements in common.
However we are not able to prove it. Following the proof of Theorem <ref>, an independent set I of the maximal size satisfies the equality in (<ref>) and thus χ_I belongs to the zeroth, the first and the second eigenspaces.
Main obstacle in our attempt is a relatively complicated structure of the second eigenspace.







  
Acknowledgements. The research is supported by Bulgarian NSF grant KP-06-N72/6-2023.	




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 Paper Title: The Aldous--Lyons Conjecture I: Subgroup Tests.












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L. Bowen]Lewis Bowen
Lewis BowenDepartment of Mathematics 1 University Station C1200 University of Texas at AustinAustin, TX, 78712 USA.
lpbowen@math.utexas.edu

M. Chapman]Michael Chapman
Michael ChapmanCourant Institute of Mathematical SciencesNew York University,251 Mercer St, New York, NY 10012, USA.
mc9578@nyu.edu

A. Lubotzky]Alexander Lubotzky
Alexander LubotzkyWeizmann institute of ScienceRehovot, Israel.
alex.lubotzky@mail.huji.ac.il

T. Vidick]Thomas Vidick
Thomas VidickWeizmann institute of ScienceRehovot, Israel.
thomas.vidick@weizmann.ac.il


§ ABSTRACT 
    This paper, and its companion <cit.>, are devoted to  a negative resolution of the Aldous–Lyons Conjecture Aldous_Lyons_Conj,Aldous–Lyons_conj_blogpost. This conjecture, originated in probability theory, is well known (cf. <cit.>) to be equivalent to the statement that every invariant random subgroup of the free group is co-sofic. We disprove this last statement.
    

    
   In this part we introduce subgroup tests. These tests are finite distributions over continuous functions from the space of subgroups of the free group to {0,1}. Subgroup tests provide a general framework in which one can study invariant random subgroups of the free group. Classical notions such as group soficity and  group stability arise naturally in this framework. By the correspondence between subgroups of the free group and Schreier graphs, one can view subgroup tests as a property testing model for certain edge-labeled graphs. This correspondence  also provides the connection to random networks.
    
    Subgroup tests have values, which are their asymptotic optimal expectations when integrated against co-sofic invariant random subgroups.  Our first main result is that, if every invariant random subgroup of the free group is co-sofic, then one can  approximate the value of a subgroup test up to any positive additive constant. 

    Our second main result is an essentially value preserving correspondence between certain non-local games and subgroup tests. By composing this correspondence with a stronger variant of the reduction in ^*= <cit.>,  proved in the companion paper <cit.>, we deduce that approximating the sofic value of a subgroup test  is as hard as the Halting Problem, and in particular, undecidable. The combination of our two main results proves the existence of non co-sofic invariant random subgroups of the free group. 

    

The Aldous–Lyons Conjecture I: Subgroup Tests
    [
    August 5, 2024
=============================================





§ INTRODUCTION

In their seminal paper <cit.>, Aldous and Lyons ask whether every unimodular network is sofic. Conditional on a certain result which will be addressed in a companion paper <cit.>, we prove the answer is `no'.
This conjecture, which originated in probability theory, can also be expressed in the language of invariant random subgroups, and this is the one we will use in this paper.
We explain and motivate this conjecture before discussing our approach. The formal definitions and results will start in  Section <ref>.



 §.§ The Aldous–Lyons Conjecture




  §.§.§ What is soficity?

Roughly speaking, a countable group Γ is sofic if it admits a sequence of partial actions on finite sets which approximates the action of Γ on itself by left-translations. If Γ=⟨ S | R⟩ is finitely presented then this is equivalent to the existence of a sequence of finite graphs which converge to the Cayley graph of Γ in the sense of Benjamini–Schramm, which is a kind of local-on-average convergence.

To be more precise, a rooted graph is a pair (G,o) where G=(V,E) is a graph and o ∈ V is a distinguished vertex called the root. A random rooted graph (G,o) is sofic if there exists a sequence (G_i)_i ∈ of finite graphs such that, if o_i is a uniformly random vertex of G_i, then (G_i,o_i) converges to (G,o) in distribution (this means: for every r>0, the radius r neighborhood of the root in G_i converges in distribution to the radius r neighborhood of the root in G). This notion of convergence is due to Benjamini and Schramm <cit.>: They proved that if each finite graph G_i is planar and there is a uniform degree bound, then the limit (G,o) is almost surely recurrent. 

The notions of Benjamini–Schramm convergence and soficity naturally generalize from graphs to labeled graphs or networks <cit.>, simplicial complexes <cit.>, manifolds <cit.>, measured equivalence relations <cit.>, measured groupoids 
<cit.> and most generally, measured metric spaces <cit.>. 

Again, a finitely presented group is sofic if one of its Cayley graphs is sofic.[If one of the Cayley graphs is sofic, then all those that use finitely many generators are.]  For example, amenable groups are sofic. In fact, the Cayley graph of an amenable group admits a sequence of finite sub-graphs in which the isoperimetric ratio (number of boundary vertices to number of vertices) tends to zero, which implies this sequence Benjamini–Schramm converges to the Cayley graph.  Also, residually finite groups are sofic because they admit sequences of finite quotient groups whose Cayley graphs approximate the Cayley graph of the given group. 

Gromov implicitly introduced sofic groups in  <cit.>. He was motivated by Gottshalk's Conjecture: If Γ is a countable group and k ∈, then any Γ-equivariant continuous map ϕ:[k]^Γ→ [k]^Γ which is injective is necessarily surjective. This is obviously true when Γ is finite. Gromov proved that it holds when Γ is sofic. Benjy Weiss gave another proof and coined the term `sofic' from the Hebrew word for finite <cit.>.



      The soficity property has been useful in obtaining positive results about L^2 and spectral theory invariants <cit.>, group rings MR2089244, MR2417890, invariant couplings of random fields <cit.>, and in constructing dynamical invariants of groups actions MR2552252, MR2854085, MR3077882, MR3132735, MR3993930. See Cap_Lup_Sofic_Hyperlinear_book, MR3821628 for more background on sofic groups. The basic idea is roughly the same in each case: One transfers properties of the finite approximating objects to properties of the limit.  

      It is a major open problem to determine whether all countable groups are sofic. We do not directly address this problem because the class of objects we study, described next, is more general than the class of groups.

 





  §.§.§ What is unimodularity?






The main tool for analyzing Benjamini–Schramm limits is called the Mass Transport Principle which makes precise the intuitive notion of statistical homogeneity. To explain, a doubly-rooted graph is an ordered triple (G,o_1,o_2), where o_1,o_2 are vertices of G. A transport function is a function f, satisfying some measurability condition, which takes as input a doubly rooted graph and outputs a non-negative real number. The interpretation is that f(G,o_1,o_2) is the amount of mass the first root o_1 sends to the second root o_2. So if (G,o) is a random rooted graph then _(G,o)[∑_x ∈ V(G) f(G,o,x)] is the average amount of mass sent out of the root. Symmetrically, _(G,o)[∑_x ∈ V(G) f(G,x,o)] is the average amount of mass sent into the root. If these two quantities are equal for every transport function f, then we say (G,o) satisfies the Mass Transport Principle. Equivalently, we say it is unimodular (this is the term used by Aldous and Lyons in <cit.>). Alternatively, unimodularity can be formulated in terms of graphings or as invariance with respect to the root-changing equivalence relation on the space of rooted graphs. The former arises from the ergodic theory of measured equivalence relations (e.g. <cit.>). The latter point-of-view was introduced in <cit.> and further developed in <cit.>.


The term unimodular comes from the following special case. Suppose G is a connected transitive graph; this means that its automorphism group Aut(G) acts transitively on its vertex set. Then (G,o) is unimodular if and only if the automorphism group Aut(G) is unimodular in the sense that its left and right Haar measures agree.











The Mass Transport Principle arose in Häggström's study of percolation clusters on trees <cit.>. In fact, it is an exercise to show that if (G,o) is unimodular and p ∈ [0,1] is given, then the Bernoulli p-percolation cluster containing o is also unimodular. Unimodularity was further developed in <cit.> (see also <cit.> where it was used to construct a version of the Euler characteristic for unimodular planar maps).

It is straightforward to verify that (1) unimodularity is closed under weak limits and (2) if G is a finite graph and o is a uniformly random vertex of G then (G,o) is unimodular. It follows from these observations that soficity implies unimodularity. Question 10.1 of <cit.> asks whether unimodularity implies soficity.

As above, unimodularity has been generalized to random rooted labeled graphs, simplicial complexes, manifolds and so on. Benjamini and Schramm note that many results which are known to hold in the deterministic setting of unimodular transitive graphs can be generalized to  unimodular random graphs <cit.>. Aldous and Lyons demonstrate this with results about random walks, amenability, ends of graphs, spanning forests, percolation, and so on <cit.>.



  §.§.§ What is an invariant random subgroup?


As a first step towards our goal, let us describe the algebraic formulation of the Aldous–Lyons Conjecture in terms of invariant random subgroups, which were introduced in <cit.>, <cit.>, <cit.> (and implicitly in <cit.>). The concept of invariant random subgroups has been profitably studied in the context of locally compact groups (e.g. stuck1994stabilizers, MR3664810,Gelander_ICM2018). 
However, we will restrict our focus to the case of most relevance to this paper: finitely generated groups. 
Already in this discrete setup, invariant random subgroups play an important role in graph convergence <cit.>, stability properties of groups <cit.> and the analysis of dynamical systems stuck1994stabilizers, MR3193754 for example.

So, fix a finitely generated group Γ. An invariant random subgroup (or IRS) of Γ is a random variable taking values in the space of all subgroups of Γ, whose law is invariant under the conjugation action of Γ. For example, every normal subgroup of Γ is an IRS. More generally, if H ≤Γ has only finitely many conjugates, then a uniformly random sample of its conjugates is an IRS. As finite index subgroups have only finitely many conjugates, they induce  IRSs which are called elementary. To generate more examples, one can take convex combinations and weak* limits of elementary IRSs, and the IRSs that arise this way are called co-sofic. It turns out that for some groups every IRS is co-sofic, and for others there are non co-sofic IRSs (cf. <cit.>). The Aldous–Lyons Conjecture can be formulated as follows: 


    Are all IRSs of a (non-commutative) free group co-sofic?

 
For the rest of the paper, this is the formulation that we tackle, and it is known — and we explain it in the following paragraphs — to be equivalent to the probability theoretic formulation on Benjamini–Schramm limits of finite graphs.

The law (or distribution) of an IRS is a Borel probability measure μ on 𝔰𝔲𝔟(Γ), the space of subgroups of Γ. We let (Γ) denote the space of all such measures. By abuse of language, we say that μ is an IRS if μ∈(Γ).
To see the relation between the two formulations of the Aldous–Lyons Conjecture, note that if H is a subgroup of Γ, then the Schreier coset graph of _H\^Γ with respect to some generating set S is a rooted (directed, edge-labeled) graph, with the root being the coset of the identity.  This gives a map from the space  of subgroups 𝔰𝔲𝔟(Γ) to the space of isomorphism classes of rooted (edge-labeled, directed) graphs. 

If μ∈(Γ) then μ pushes forward under this map to a unimodular measure <cit.>. 
 Moreover, if μ is elementary, then its pushforward is induced by uniformly choosing a root of some finite connected graph. Therefore, if μ is co-sofic, then its pushforward is sofic.

Now, consider the special case in which Γ=(S) is the free group generated by a finite set S. Suppose μ is an invariant random subgroup of Γ, and that the (random) Schreier coset graph of μ is sofic (in the directed, edge-labeled rooted graph category). This means there exists a sequence of finite directed, edge-labeled graphs G_i which Benjamini–Schramm converge to the (random) Schreier coset graph of μ. After perturbing the given sequence if necessary, we may assume that each G_i is itself a Schreier coset graph with respect to some finite-index subgroup H_i of (S) — the existence of such perturbations is exactly where the freeness of the group (S) plays a role. Moreover, if H_i is a uniformly random conjugate of H_i, then the distribution of H_i converges to μ. Therefore, a positive solution to the Aldous–Lyons Conjecture implies that every IRS of (S) is co-sofic.[There is a standard “decoration of graphs” technique that allows one to show that the directed, edge labeled (with finitely many labels) version of the Aldous–Lyons conjecture is equivalent to the non-directed, non-edge labeled version. As Bálint Virág pointed to us, the same technique is used, e.g., to show that every group is the automorphism group of some graph. As this is standard, we do not elaborate on it anymore.]







 §.§ Our approach



  §.§.§ What is a subgroup test?

The set () of all IRSs of the free group and the set _() of co-sofic IRSs of the free group are both convex. Thus, the Aldous–Lyons Conjecture is asking whether certain convex sets can be separated. A natural way to distinguish between two convex sets is to use (continuous) linear functionals. So, in order to separate the above, one should search for a rich enough collection of functionals. Subgroup tests and their values will play this exact role.  

A challenge is a continuous map from () to {0,1}. A subgroup test  is a probability distribution on a finite set of challenges. One can integrate any subgroup test against any probability measure over (). If π is in (), then this integral is called the value of the strategy π against the test .

 Many natural and well studied properties of groups, specifically soficity and pointwise permutation stability, can be formulated as properties of subgroup tests. 

The terminology of “tests” and “values” originates in the field of interactive proofs in computer science. 
An interactive proof designates an interaction between two entities, a “verifier” and a “prover”. The goal of the prover is to convince the verifier of the validity of a certain claim, and the goal of the verifier is to test the prover so that it  accepts the interaction only if the claim  is indeed correct. 
Examples of “claims” studied in computer science are the 3-colorability of a given input graph, or that a Turing machine whose description is passed as input halts.
In our approach, the methods of the verifier and the prover are restricted. The prover is required to sample an element of (). 

The sampling method itself is restricted to some sub-class of IRSs  (e.g., elementary IRSs); the choice of distribution according which to sample is the only degree of freedom the prover has, and is called its strategy. The verifier gets access to (the indicator function of) the prover's sampled subgroup H, and it makes a decision whether to accept or reject this subgroup. 
To decide, the verifier holds some finite set K ⊆ — known beforehand to the prover — and according to which combinations of elements of K are contained in H, it accepts or rejects; the rules that control which combinations of elements result in acceptance and which combinations result in rejection are also know  to the prover beforehand. The value of a strategy in a game is the probability, taken over the prover's sampling and the verifier's probabilistic choices, that the verifier makes the decision to accept when interacting with a prover using this strategy. 

In spite of these restrictions, it is possible to verify hard computational problems in this setup.[For the expert, we note that restrictions on the  prover and verifier can affect both the completeness and soundness properties of a proof system. Hence, a priori, they may either lower or raise the complexity of the associated class of interactive proofs compared to classical single-prover interactive proofs.] For example, it is not difficult to design a subgroup test such that there is a value-1 strategy for this test if and only if a given graph is 3-colorable. Jumping ahead, the correct analogy is with the theory of multiprover interactive proofs, and even more specifically interactive proofs with quantum provers sharing entanglement, also known as non-local games. We explain below this connection, which plays an essential role in the second part of this paper.





  §.§.§ What is a non-local game?

Morally, non-local games originated in Bell's resolution <cit.> of the Einstein–Podolsky–Rosen paradox <cit.>. Einstein was famously concerned by the ability of space-time separated particles to correlate in ways that might  violate relativity, a possibility that seemed to be suggested by the mathematical modeling of quantum mechanics. Einstein argued for a local hidden variable theory — namely, that the particles did share some information beforehand,  which allowed them to correlate — and that physicists should search for this hidden information.
Bell provided a thought experiment, that in modern jargon is an instance of a non-local game, which proves that the kinds of correlations that arise from isolated quantum mechanical systems are intrinsically different from those that could be generated in any hidden variable model. 
Bell's work lay the foundation for the subsequent design, and execution, of experiments which verify the existence of quantum entanglement (and thus refute Einstein's approach; see also <cit.>). The 2022 Nobel prize in physics was awarded to Aspect, Clauser and Zeilinger, partly for performing non-local games as experiments and verifying that the winning statistics in the games exceeds what local hidden variable models allow.


A correlation is a function p A× A× X× X→ [0,1], where X and A are finite sets, such that for every pair x,y∈ X the function p(·,·|x,y) is a probability distribution over A× A — the quantity p(a,b|x,y) should be thought of as the answer to “what is the probability a and b are provided as answers given that x and y were asked as questions?”; this cryptic phrase will soon be clarified.  A correlation  p is said to be deterministic if there are two functions f_1,f_2 X→ A such that p(a,b|x,y)=1 only if f_1(x)=a and f_2(y)=b.
The convex hull   C_c of  deterministic correlations is the set of local or  classical correlations.[An example of a non-local correlation is p(a,b|x,y)=1 only when a=y and b=x. Intuitively, this is because this correlation implies signaling from one system to the other.] 
On the other hand, the  quantum correlations  C_q are those that can be generated by performing quantum measurements on a (finite-dimensional) bipartite physical system, where the measurement on one part depends only on x and generates a, while  the measurement on the other part depends only on y and generates b. For a formal description, see Remark <ref>. The set C_q is convex, but interestingly it is not closed <cit.>.

A non-local game  is specified by a probability distribution μ on the set X×X, and a decision function D:X× X× A× A→{0,1}.[As can be seen in Section <ref>, our definition of non-local games is slightly different. This difference is mainly cosmetic, and is driven by our motivation to relate non-local games to subgroup tests.] 
The interpretation of this combinatorial object  as a “game” comes from thinking of it (similarly to subgroup tests) as an interactive proof, but this time with two provers, a la <cit.>. 
The common dramatization goes as follows: Two provers are spatially separated. The verifier samples a pair of “questions” x,y∼μ, and “asks”  one prover x and   the other y. The provers then apply some local procedure to choose their “answers” a and b respectively. Finally, the verifier accepts if D(x,y,a,b)=1 and rejects otherwise.  

Since the process by which the provers generate their answers is hidden to us, we can only “observe” (samples from) the distribution of answers given questions, namely, the underlying correlation. So, the correlation can be thought of as the “strategy” used by the provers (similar to the way IRSs are seen as strategies for the prover in a subgroup test).  Bell's separation of C_c from C_q amounts to devising a game that is “easy” for players that use quantum correlations, yet “hard” for players that use only classical ones. 

Now, there is more than one candidate mathematical model for entanglement in quantum mechanics. A slightly generalized model, suggested by Tsirelson <cit.>,  gives birth to a set of correlations known as quantum commuting correlations C_qc. This set is closed and contains the quantum correlations.  Tsirelson famously asked whether the closure of C_q is equal to C_qc <cit.>. At this point, we hope the type of problem already resonates with the reader, as this is again a separation of convex sets type of problem.[Connes' embedding problem (CEP) is also a separation of convex sets type of problem, specifically, separating all  characters of the free group from those that are limits of finite dimensional characters. This problem was shown <cit.> to be equivalent to Tsirelson's problem, and also has close connections with invariant random subgroups — as every IRS induces a character. We omit discussions on the relations between this work and CEP from the text, as they are not particularly helpful for understanding our approach.]



  §.§.§ Harnessing undecidability

It was known that if  the quantum correlations are dense in the quantum commuting correlations, then the complexity class of multiprover interactive proofs with entangled provers (^*) contains only decidable languages.
By proving that the Halting Problem, which is undecidable, is in ^*, the authors of <cit.> were able to resolve Tsirelson's problem in the negative.


In this paper a similar path is followed: Rather than directly find a functional that separates the co-sofic and general IRSs of the free group, we describe a computational problem — approximating the sofic value of a subgroup test — whose undecidability implies a negative resolution of the Aldous–Lyons conjecture.  Our approach can be described in two steps: First, show that a positive solution to the Aldous–Lyons Conjecture implies that approximating the sofic value of a subgroup test is decidable. Second, prove that this computational task is undecidable.

Elaborating on the first,  given a subgroup test , we prove that its optimal value against co-sofic IRSs — which we call the sofic value of  — can be approximated from below, and its optimal value against any IRS — which we call the ergodic value of  — can be approximated from above.  Thus, if the Aldous–Lyons Conjecture has a positive answer, then the sofic and ergodic value of a test always agree, and using the aforementioned approximation procedures they can be calculated to any  predetermined accuracy. 

Elaborating on the second, we provide a mechanism for translating certain non-local games (see Section <ref>), which we call tailored non-local games, to subgroup tests. This translation is essentially value preserving. Namely, if the non-local game had a (certain kind) of perfect quantum strategy — which we call a Z-aligned permutation strategy that commutes along edges — then the analogous subgroup test has a perfect (co-sofic) strategy. In the other direction, if the subgroup test has an almost perfect (co-sofic) strategy, then the non-local game has an almost perfect strategy. Hence, a certain strengthening of the reduction in <cit.>, which states that small additive constant approximations to the value of tailored non-local game are undecidable (Theorem <ref>), implies that the sofic value of a game cannot be approximated to arbitrary precision. Thus, the Aldous–Lyons Conjecture has a negative solution.




The rest of the introduction provides a deeper dive into the  definitions, results and ideas   of this paper.




    

 §.§ Our approach





§ INTRODUCTION

Probabilistic methods for studying algebraic, geometric and combinatorial objects have proved themselves extremely powerful (cf. alon2016probabilistic,tenenbaum2015introduction). To better understand a collection of objects, you define a sampling scheme on it and analyze the expected behaviours of the samples. Invariant random subgroups, defined in <cit.>, are a flexible tool of this exact kind: They allow one to study lattices in Lie groups <cit.>, graph convergence <cit.>, stability properties of groups <cit.> and various dynamical systems <cit.>.

Given a countable group Γ, let (Γ) be the collection of all subgroups of Γ. An invariant random subgroup (IRS) is a Borel probability distribution over (Γ) which is invariant under conjugation. The most straightforward example of an IRS is the Dirac measure concentrated on a normal subgroup. Another basic example is the uniform distribution over the conjugation orbit of a finite index subgroup. We call such an IRS simple. To generate more examples, one can take convex combinations and weak* limits of the simple IRSs. The IRSs that arise this way are called co-sofic. It turns out that for some groups every IRS is co-sofic, and for others there are non co-sofic IRSs (cf. <cit.>). A major open problem in this field, due to Aldous and Lyons (Conjecture <ref>), asks  which is true for a (non-commutative) free group? In this paper (and its companion <cit.>), we prove that non-commutative free  groups have non co-sofic IRSs. 

The set of all IRSs of the free group, and the set of co-sofic IRSs of the free group, are both convex. Thus, the Aldous–Lyons conjecture is asking whether certain convex sets can be separated. The natural way to distinguish between convex sets is by using linear functionals. Thus, one should search for a rich collection of functionals if they seek to separate the above. Subgroup tests and their values will play this role.

Though it is tempting to try and find a specific functional that separates the co-sofic and general IRSs  of a free group, we suggest another approach in the spirit of the recent resolution of Tsirelson's problem <cit.>.  Tsirelson's problem is also a separation problem of convex sets, the quantum and quantum commuting correlations. 
It was known that if  the quantum correlations are dense in the quantum commuting correlations, then the complexity class of Multi-prover Interactive Proofs with Entangled Provers (MIP^*) contains only decidable languages.
Thus, by proving that the Halting Problem is in MIP*, they were able to resolve Tsirelson's problem.
In this paper we construct a similar setup, namely, define a complexity class whose containment of undecidable languages will imply the existence of non co-sofic IRSs in the free group. 
For those familiar with multi prover interactive proofs, we remark that  subgroup tests  are analogous to non-local games. They are combinatorial objects that test probability distributions over subsets of the free group. Many natural and well studied properties of groups, specifically soficity and pointwise permutation stability, can be formulated as properties of subgroup tests. 


In the rest of the introduction we give brief definitions of the relevant notions, and overview the main results of this paper.





 §.§ Subgroup Tests
 
Let S be a finite set, =(S) be the free group with basis S, () the collection of subgroups of  and Prob(sub(ℱ)) the set of all Borel probability measures on sub(ℱ). 
A challenge  is a pair (K;D) such that K is a finite subset of , and D is a function from subsets of K to {0,1}. A subset A⊆ passes the challenge (K;D) if D(A∩ K)=1, and fails it otherwise.
A test  is a finite collection of challenges {(K_i;D_i)}_i∈ Q, where Q is a finite index set, together with a probability distribution μ over the set Q. 
A strategy is a (Borel) probability distribution over subgroups of the free group, namely an element π∈( ()). 
One can run the test  against π, as follows. 

First the verifier samples i∈ Q according to μ. Next, the prover samples a subgroup  H according to π. Finally, the verifier makes the decision to accept if H passes the i^ th challenge (K_i;D_i), and reject if H fails the i^ th challenge.






The value of using the strategy π against the test  is its acceptance probability, namely

    (,π)=_H∼π_i∼μ[D_i(H∩ K_i)].

For a fixed , the value is a continuous linear functional from probability distributions over subgroups of  to ℝ. 
Hence, it enables us to study convex subsets of (()) by their optimal value against a given test. 


    As mentioned earlier, a challenge is, essentially, a continuous map from the space of subgroups of  to {0,1}. One can extract from any such continuous map  a challenge (K;D) that describes it and vice versa. Therefore, a test is a convex combination of continuous maps from () to {0,1}, and the value is integration of this map against a (probability) measure. 







 §.§ Invariant random subgroups


As sub(ℱ) inherits the product topology
from the power set {0,1}^ℱ,  it is compact.
Every action of a group Γ on a set X extends naturally to an action of Γ on the power set {0,1}^X by 

    ∀γ∈Γ,A⊆ X   γ.A={γ.a | a∈ A} .

Hence, the conjugation action of ℱ on itself is inherited by {0,1}^. Also, ()⊆{0,1}^ is preserved by this action. Hence, the conjugation action extends  to its power  set {0,1}^() by 

    ∀ w∈ℱ, A⊆sub(ℱ)    w. A={wHw^-1| H∈ A} .

 The conjugation action of ℱ further extends to Prob(sub(ℱ)):
For every Borel set B⊆() and μ∈ Prob(sub(ℱ)), we have

    (w.μ)(B)=μ(w^-1.B) .
 



The set of invariant random subgroups IRS(ℱ)
consists of all probability measures in Prob(sub(ℱ))
that are invariant under the action of ℱ. 


The ergodic value of a test  is the best performance of a strategy π∈() against it, namely

    _()=sup{(,π) |π∈()} .


The extremal points in the convex set  (Γ) are commonly referred to as ergodic IRSs.
    As we take the supremum of a linear functional over a convex compact set, the optimum is obtained on an extremal point, which explains the phrase ergodic value. 




 §.§ Finitely described invariant random subgroups
 
Let X be a finite set, and let (X) be the symmetric group acting on X. Every map σ S→(X) extends uniquely to an action of =ℱ(S) on X. 
For a vertex x∈ X, we can define its stabilizer to be 
    Stab(σ,x)={ w∈ℱ|σ(w).x=x} ,

which is a subgroup of ℱ.
Hence, we can associate with σ S→(X) an IRS of  via the following sampling procedure: 
(1) Choose x∈ X uniformly at random. (2)
  Output Stab(σ,x).
We denote by Φ the map from finite actions of  to () defined by this procedure, namely

    Φ(σ)=_x∈ X[ 1_ Stab(σ,x)] ,

where 1_H is the Dirac measure concentrated on the subgroup H and _x∈ X is the expectation according to the uniform measure over X. The set of finitely described invariant random subgroups _() is the image of the map Φ in (). 

The sofic value of a test  is the (asymptotic) best performance of a strategy π∈_() against it, namely

    _()=sup {(,Φ(σ)) | |X|<∞, σ S→(X)} .




The name sofic value is appropriate, as demonstrated in Section <ref>. Furthermore, we often abuse notation and use (,σ) instead of (,Φ(σ)), even though the action itself is not an IRS.



Since _()⊆(), we have _()≤_() for every test . The Aldous–Lyons conjecture <cit.> can be formulated as follows:
  Is  IRS_(ℱ)
 weak* dense in IRS(ℱ)?


In their paper <cit.>, Aldous and Lyons ask this as a question. In the passing of time, the positive form of this problem, namely that _() is dense in (), received the name The Aldous–Lyons Conjecture (cf. <cit.>). Thus, we use  the more common “Conjecture” phrasing instead of Problem. 

The algebraic formulation we  provided earlier, which can also be found  in Section 6 of <cit.>, may seem different from the above. The question was  whether every invariant random subgroup of  is co-sofic, where co-sofic means an IRS which is the weak* limit of convex combinations of uniform distributions over finite index subgroups. Since the finitely described IRSs are dense in the co-sofic ones, the formulation in Conjecture <ref> is equivalent to this standard (algebraic) formulation.



Lastly, the term co-sofic is closely related to the better known term of a sofic group (cf. Definition <ref>).
A finitely presented group ℱ/N is sofic if and only if the Dirac measure concentrated on N is a co-sofic IRS (cf. Proposition 6.1 in <cit.>).
      Thus, a positive solution to the Aldous–Lyons Conjecture <ref> implies in particular that every group is sofic.[The way we formulate it, a positive solution to the Aldous–Lyons Conjecture only implies that all finitely presented groups are sofic. It is standard to deduce it for all groups from the finitely presented ones <cit.>.  ] 


      The soficity property has been useful in obtaining positive results about group rings <cit.>, invariant couplings of random fields <cit.>, and invariants of groups actions <cit.>. See <cit.> for more background on sofic groups.

    




 
Since (,·) is a continuous functional, a positive solution to the Aldous–Lyons Conjecture <ref> implies that  for every test ,  _()=_().



    Due to the relation between subgroups of the free group and Schreier graphs, one can view subgroup tests as a new property testing model for (edge-labeled) graphs:
Let X be a finite set and σ S→ X  a transitive action (the non-transitive case is a convex combination of transitive ones). For x∈ X, the stabilizer Stab(σ,x) is exactly the set of labeled paths in the Schreier graph of (S) with respect to Stab(σ,x) and S that begin and end at x, namely, closed paths originating from x. Thus, when the   finitely described IRS Φ(σ) is put against a test , we actually test the aforementioned Schreier graph  in the following way:
First, choose a vertex x∈ X uniformly at random and sample i∼μ.
  List the paths from K_i and for each one check whether it is closed or open.  According to this check,  decide using D_i whether to accept or reject.





 §.§ Decidability of approximating the sofic value
 

    We briefly recall basic notions from computability and complexity theory.
Let Σ be a finite set that will play the role of an alphabet.
Let Σ^* be the free monoid generated by Σ, namely
the set of all finite words in Σ. 
A language is a subset L⊆Σ^*. Every language
L defines a decision problem: given w∈Σ^*, decide
whether w∈ L.
A promise language is a pair L_yes,L_no⊆Σ^*
such that L_yes∩ L_no=∅. A promise language (L_yes,L_no)
defines a promise decision problem: given w∈ L_yes∪ L_no,
decide whether w∈ L_yes. A promise language (L_yes,L_no) is decidable if there exists a Turing Machine (TM for short) M satisfying the following: If w∈ L_yes∪ L_no is given to M as input, then M halts and outputs a single bit M(w) satisfying

    M(w)=1     w∈ L_yes,
    
        M(w)=0     w∈ L_no.

Let L^1 and L^2 be two (promise) languages.  We say that L^1 reduces to L^2 if there exists a time Turing machine M for which: If σ∈ L^1_yes (respectively,  σ∈ L^1_no) is taken as the input of M, then M halts and the output M(σ) is in L^2_yes (respectively, M halts and the output  M(σ) in L^2_no).
Thus, if L^2 is decidable then L^1 is also decidable.







If the  challenge distribution μ in a subgroup test  is rational, then  can be encoded as a finite bit string. There are many (non-equivalent) ways of doing so.  
Assume we fixed some encoding for tests with rational challenge distributions such that, given the encoding, all underlying combinatorial data of the test can be calculated in finite time. 
Namely, the  set of generators S, the indexing set of challenges Q, the collection of challenges {(K_i;D_i)}_i∈ Q and the distribution μ can all be read from the encoding in finite time.
 

For every fixed encoding of tests such that all combinatorial data of the test can be read from it in finite time:

    
  * There is a Turing machine M_1 that takes as an input (the encoding of) a test , and outputs an infinite sequence of non-decreasing numbers (α_t)_t=1^∞⊆ [0,1]  such that lim_t→∞α_t=_().
    
  * There is a Turing machine M_2 that takes as an input (the encoding of) a test , and outputs an infinite sequence of non-increasing numbers (β_t)_t=1^∞⊆ [0,1]  such that lim_t→∞β_t=_().



    Clause (2) of Theorem <ref> should be compared to the Navascués–Pironio–Acín (NPA) hierarchy in the study of non-local games <cit.>. See also the introduction of <cit.>.



    If the Aldous–Lyons Conjecture <ref> has a positive solution, then for every θ>0, there is a Turing machine which accepts a subgroup test  as input and outputs its sofic value _() up to an additive error of at most θ.






























 §.§ Tailored non-local games and their associated subgroup tests





As described before, the standard definition of  a (non-local) game consists of two finite sets, a questions set X and an answers set A, together with a probability distribution μ over X× X and a decision function D X× X× A× A→{0,1}. In this paper, we use a slightly modified definition, that makes the connection with subgroup tests clearer. Before diving into our alternative definition, note first that the distribution μ defines a graph structure on X, by associating edges with its support. Furthermore, as the exact choice of A is irrelevant, we can assume it is the set of all bit strings up to some length Λ.

A (synchronous, non-local) game consists of a graph  G=(V,E), a distribution μ over E,
a length function ℓ V→ℕ,  formal sets of generators S_x={^x,i| 1≤ i≤ℓ(x)} — the union of which is denoted by S=⋃_x∈ VS_x — and decision functions D_xy{0,1}^S_xy→{0,1} for every edge xy∈ E, where S_xy=S_x∪ S_y.[So, the main difference in our definition, is that the decision function D rejects automatically answers that are not of the appropriate length according to ℓ.]
     A (synchronous, quantum) strategy for  is a map ρ⋃_x∈ VS_x→ U(n), where the images are involutions, and ρ(S_x) commute for every fixed x∈ V.[This is again not the standard definition of a quantum strategy. Unpacking our definition, ρ associates an observable (Hermitian operator on a finite-dimensional Hilbert space that squares to the identity) with each bit of the answer when asked for x∈ V. Since the answer must be totally measured, all the observables associated with a specific vertex must commute. The fact there is a single  map ρ, and not one for each prover, is the synchronicity of this strategy. ]
Such a strategy defines for every xy∈ E a probability distribution over maps γ S_xy→{0,1} in the following way: 
For every ∈ S, let  ^_0 be the projection on the (+1)-eigenspace of ρ(), and ^_1  the projection on its (-1)-eigenspace. 
Furthermore, let τ be the dimension normalized trace on n× n matrices.  
Then, the probability that γ S_xy→{0,1} is sampled according to ρ is by definition 

    τ(∏_∈ S_x^_γ()∏_∈ S_y^_γ()) .

A map γ sampled that way is said to be sampled according to ρ, and we denote it by γ∼ρ.[Note that this sampling depends on the chosen edge xy∈ E. Namely, though implicit in the notation, γ is defined from S_xy for some edge xy∈ E. It is important to note that γ is not globally defined on all of S.]
    Relating to standard notations, we remark that by letting a=γ|_S_x and b=γ|_S_y, the correlation p(a,b|x,y) induced by the quantum strategy  ρ (as in Remark <ref>) agrees with   formula (<ref>).

    In a similar manner to subgroup tests, one can run the game  against the strategy ρ: 
    
    First, the verifier samples an edge xy∈ E according to μ. Then, the prover samples γ S_xy→{0,1} according to ρ. Finally, the verifier accepts if D_xy(γ)=1, and rejects otherwise.
   
   
   
   
   
 
  As for subgroup tests, the  value of the strategy ρ against the non-local game  is its acceptance probability when ran against the game. Namely,

    (,ρ)=_xy∼μ_γ∼ρ[D_xy(γ)] .

 The (synchronous) quantum value of , ^*(), is the supremum of (,ρ) over all possible quantum strategies against .


The main result in <cit.> (recalled in detail in Theorem <ref> herein) is a reduction from the Halting Problem to approximating  the quantum value of a game.
Our main contribution in the second half of this paper (Sections <ref>,<ref> and <ref>) is a mapping from a specific sub-class of non-local games — which we term tailored non-local games

and describe further below — to subgroup tests, that  (almost) preserves their values.


    There is a transformation that takes as input a tailored non-local game (see Definition <ref>) and outputs a subgroup test (which we call the associated synchronous subgroup test, see Definition  <ref>), such that:
    
        
  * If the game had a perfect (i.e., a value 1) Z-aligned, commuting along edges, permutation strategy (see Definitions <ref> and <ref>), then the associated subgroup test has a perfect strategy. 
        
  * Every almost perfect finitely described strategy for the associated subgroup test can be transformed into an almost perfect quantum strategy to the original tailored non-local game. 
    




Informally, a tailored game is one where the decision function D_xy takes the following form. First, D_xy examines the value taken by γ on a subset of the generators S_x∪ S_y (we refer to these as readable variables). Based on this, D_xy determines a collection of linear equations on the entire string γ. Finally, D_xy accepts if and only if all these equations are satisfied. This form for the game can be seen as a generalization of binary linear constraint system games (Example <ref>), which corresponds to the case where the set of readable variables is empty. 

A permutation strategy, as its name suggests, is a quantum strategy which is induced by permutations. Such a strategy is said Z-aligned when the observables associated with the readable variables (i.e. the bits of the answer γ that control the parity checks in the tailored game) are all diagonal matrices (with respect to some natural basis). The restriction to Z-aligned strategies is crucial for item (1) in Theorem <ref> above to hold. 

In a companion paper <cit.>,  we prove that the reduction of the Halting Problem to approximating the quantum value of a game presented in <cit.> can be modified such that the resulting non-local game is tailored and satisfies the required stronger assumptions (Theorem <ref>): If the input Turing machine halts, then the game has a perfect Z-aligned permutation strategy that commutes along edges, and if the input Turing machine does not halt, then the value of the game is bounded from above by 1/2. Therefore, together with our mapping from non-local games to subgroup tests, we deduce that approximating the sofic value of a game is undecidable. In turn, by Corollary <ref>, 
  the Aldous–Lyons Conjecture <ref> has a negative solution.


    In <cit.>, the authors were able to  specify a non-local game whose quantum value is bounded above by 1/2 but has a perfect quantum commuting strategy. Using the results of <cit.>, which we elaborate on further in the next remark,  this implies the existence of a finitely presented *-algebra which can be endowed with a tracial state and cannot be embedded into an ultrapower of the hyperfinite II_1 factor. This is a somewhat constructive counter example, but the state itself is only proven to exist and does not have a constructive form (at this point). By the various formulations of Connes' embedding problem, this proves that there is a non-hyperlinear character of the free group, which is the analogue of a non co-sofic IRS of the free group, but it is provided in a non-constructive form.

By applying the “same” construction as in <cit.> with the algorithm constructed in Theorem <ref> replacing the NPA hierarchy algorithm,  one can pin down an explicit subgroup test with perfect ergodic strategy but all sofic strategies bounded above by some constant (which can be approximated by following the proofs). This is the most constructive we can get in this case, namely, the specific non co-sofic IRS which is the perfect strategy for this subgroup test is proven only to exist, and is not constructed. Thus, we decided not to specify this test. 
    
    It would be very interesting to find an explicit non co-sofic IRS, namely separate between () and _() in a constructive manner.


  We comment on a natural attempt to extend the construction of <cit.> so as to directly deduce the existence of a non-sofic group, a result which would imply a negative answer to the Aldous–Lyons Conjecture. 

The idea (which is folklore, and explicitly mentioned in <cit.>) is as follows. It is well-known that to every synchronous game 𝒢 one can associate an algebra 𝒜(𝒢) such that there is a tight connection between the C^*-representation theory of this algebra and perfect  values for the game in various correlation sets <cit.>. In the special case where 𝒢 is a linear constraint system (LCS) game, there is a group  Γ such that 𝒜()  is (isomorphic to the completion of) the complex group algebra ℂ[Γ] <cit.>.[Actually, it is isomorphic to the completion of  ℂ[Γ]/⟨+1⟩, where  is a special generator in Γ and ⟨+1⟩ is the two-sided ideal generated by +1 in ℂ[Γ].] 

Now, if the reduction given in <cit.> returned only LCS games, then one would obtain the following result: there exists an LCS game with a perfect quantum commuting strategy, but no (asymptotically) perfect quantum strategy. By the connection mentioned above this implies that the group Γ associated with 𝒜()  is non-hyperlinear. As every sofic group is hyperlinear, one would thus obtain an example of a non-sofic group.

Since the reduction in <cit.> does not return LCS games, a workaround could be to find an embedding from the algebra 𝒜(𝒢), where 𝒢 is returned by <cit.>, into 𝒜(𝒢') where 𝒢' is an LCS game. Obstacles to this “black-box” approach are discussed in <cit.>. 

One can view our approach as finding a “sweet spot” that is in some sense half-way between synchronous games, which were used in <cit.>, and LCS games. This is because tailored games are a natural generalization of LCS games. In the companion paper we show that <cit.> can be adapted to return tailored games, and in the present paper we show that this suffices to resolve the Aldous–Lyons Conjecture. The existence of non-sofic groups (and non-hyperlinear groups) remains an  elusive open problem. 


  

    
Recall that a (synchronous 2-player) non-local game  consists of a (finite) graph G=(V,E), a length function ℓ V→ℕ, formal sets of generators S_x={^x,i| 1≤ i≤ℓ(x)} for every vertex x∈ V, a distribution μ over the edges E, and decision functions D_xy{0,1}^S_xy→{0,1} for every edge xy∈ E, where S_xy=S_x∪ S_y. Namely, the decision function D_xy takes as input a pair of functions α S_x→{0,1} and β S_y→{0,1} and outputs a single bit. A (synchronous quantum) strategy for  is a map ρ⋃_x∈ VS_x→ U(n), where the images are of order 2, and ρ(S_x) commute for every fixed x∈ V.   Since ρ() is of order 2, we denote by ^_0 the projection on the (+1)-eigenspace of ρ(), while ^_1 the projection on its (-1)-eigenspace. Then, the  value of  ρ against  is

    (,ρ)=_xy∼μ∑_α S_x→_2 
    β S_y→_2 D_xy(α,β)τ(∏_∈ S_x∏_∈ S_y^_α()^_β()),

where τ is the dimension normalized trace on n× n matrices. The (synchronous) quantum value of , ^*(), is the supremum over all possible strategies against . The main result of MIP*=RE <cit.> is a reduction from the halting problem to approximating  the quantum value of a game. Given a Turing machine M, they construct a game _M such that: If M halts, then _M has a strategy ρ with (_M,ρ)=1 (and in particular, ^*(_M)=1).
    And if M never halts, then ^*(_M)≤1/2.
We now define a family of tests that will allow us to leverage this result, up to some caveats, to our needs.



A synchronous test  consists of a (finite) graph G=(V,E), a length function ℓ V→ℕ, a distribution μ over the edges E, and a challenge (K_xy;D_xy) associated to every edge xy∈ E. While keeping the S_x notation from before, we assume K_xy consists of only words in S_xy∪, where  is a global generator which plays the role of negation. Regardless of the specific test, the challenges always include the following checks:

    
  * All generators are order 2, namely ^2∈ H and
    
    ∀∈ S_xy   ^2∈ H.

    
  * The special generator  is non-trivial and central, namely ∉ H and 
    
    ∀∈ S_xy   [,]∈ H,

    where [·,·] is the (multiplicative) commutator.

The similarities between the two definitions are clear. One would hope that there is a straightforward way of transforming non-local games into synchronous tests, such that the values are preserved, or can  at least be bounded by one another in an appropriate manner. If this is possible, then one can take the transformation from Turing machines M to games _M a la MIP*=RE,  transform _M to the associated synchronous test (_M), and obtain a reduction from the halting problem to SOFVAL_c,s. While this is not true in general, by restricting the types of non-local games we focus on, we are able to do just that. The terms tailored form, perfect Z-aligned permutation strategy and augmented synchronoys test will be explained in Section <ref>.

There is a map from non-local games  in tailored form to (augmented) synchronous tests (()) such that:

    
  * If  has a perfect Z-aligned permutation strategy, then the associated  test (()) has a perfect strategy.
    
  * If ^*()≤1/2, then _((()))≤ 1-δ, for some universal constant δ>0.
 


    If the reduction from Turing machines to non-local games in MIP*=RE can be done such that all games are in tailored form, and in the completeness case the outcoming game has a perfect Z-aligned permutation strategy, then SOFVAL_1,1-δ is undecidable, and thus Conjecture <ref> is false.




 §.§ Structure of the paper

 In Section <ref>, we prove  structural results about invariant random subgroups, some of which are standard and well known to experts. In Section <ref>, we  prove our first main theorem, 
 Theorem <ref>,  and draw corollaries from it. 
 In Section <ref>, we provide  several examples of subgroup tests,  inspired  by  both group theory and complexity theory.  In Section <ref>, we discuss a rigidity property of tests that generalizes group stability and which is analogous to robustness of non-local games. 
In Section <ref>, we define tailored non-local games and analyze their properties. 
In Section <ref>,  we describe the (almost) value preserving correspondence between tailored non-local games and synchronous subgroup tests, informally outlined in Theorem <ref>. This allows us in Corollary <ref> to provide a negative solution to the Aldous–Lyons Conjecture <ref>. The proof of the aforementioned value preserving correspondence, Theorem <ref>, is proved in Section <ref>. Theorem <ref>, which is the other main ingredient in our refutation, is proved in  a companion paper <cit.>. There is no open problems section, but  further research directions can be found in Remarks  <ref>, <ref> and <ref>.

For a reader who seeks the main ideas of the paper without diving too deep into the proofs, we suggest reading, in addition to the introduction, only Sections <ref> and <ref>.



 §.§ Acknowledgements



We thank Alon Dogon for reading an early draft of this paper and providing many useful improvements to the presentation. We would also like to thank Mikael de la Salle for various discussions along the way. Finally, we want to thank Peter Sarnak for organizing a seminar on “strong convergence” in Princeton and asking the second author to give a talk, as it organized his thoughts  and led to the current presentation of ideas.

Lewis Bowen is supported by NSF grant DMS-2154680.
Michael Chapman acknowledges with gratitude the Simons Society of Fellows and is supported by a grant from the Simons Foundation (N. 965535).
Alex Lubotzky is supported by the European Research Council (ERC)
under the European Union's Horizon 2020 (N. 882751), and by a research grant from the Center for New Scientists at the Weizmann Institute of Science.
Thomas Vidick is supported by a research grant from the Center for New Scientists at the
Weizmann Institute of Science, a Simons Investigator award, AFOSR Grant No. FA9550-22-1-0391, and
ERC Consolidator Grant VerNisQDevS (101086733).
 


§ GROUP THEORETIC PRELIMINARIES




 §.§ Topology (and other properties) of invariant random subgroups
[The content of this section is standard. For example, it appears in Section 3 of <cit.>.]
We first go through some properties of the space {0,1}^ (induced with the product topology). For every  K,L⊆ℱ, let 

    𝒞(K,L)={A⊆ℱ| K⊆ A, L∩ A=∅}⊆{0,1}^.
 
Then,

    
  * When K and L are finite, the set (K,L) is open and closed. Moreover, the collection 
    {(K,L)| K,L⊆, |K|,|L|<∞}
 is a basis for the topology of {0,1}^. In addition, {0,1}^ is compact.
    
  * For every w∈ℱ and subsets K,L⊆ℱ we have
    
    w.𝒞(K,L)=𝒞(wKw^-1,wLw^-1).


The following is a standard fact. A proof is added because elements of it are later used in Lemma <ref>.
 
    The set of subgroups () is closed in the collection of all subsets {0,1}^. 



    A subset A⊆ is a subgroup if it satisfies the following three conditions: 
  
    (1) _∈ A ,  
           (2)  w∈ A w^-1∈ A ,  
           (3) w,w'∈ A w· w'∈ A.

For every w,w'∈, define the following open sets:  The subsets not containing _, C_=(∅,{_}). The subsets containing w∈ but not w^-1, C_w=({w},{w^-1}). The subsets containing w,w'∈ but not w· w',
    C_w,w'=({w,w'},{w· w'}).
     Then,
     
    (⋃_w,w'∈  C_w,w')∪(⋃_w∈ C_w ) ∪ C_={0,1}^∖(),
 
     which proves that () is a closed set.






Recall the definition of (), the invariant random subgroups of the free group  (Definition <ref>).


 
Let μ,ν∈ Prob({0,1}^ℱ). Then,

    
    
  * μ=ν if and only if μ(𝒞(K,L))=ν(𝒞(K,L)) for every finite K,L⊆ℱ. 
    
  * μ is conjugation invariant if and only if for every s∈ S we have s.μ=μ. Hence, μ∈() if and only if for all s∈ S and finite K,L⊆ℱ we have μ(𝒞(K,L))=μ(s.𝒞(K,L)).



A sequence of measures μ_n∈ Prob({0,1}^ℱ) is said to weak* converge to μ if for all continuous f{0,1}^ℱ→ℝ, 

    _H∼μ_n[f(H)] _H∼μ[f(H)].

We denote weak convergence by μ_n μ. 

    In our context, there is a more “down to earth” way of viewing weak* convergence. For every finite subset A⊆, and every  probability measure μ∈({0,1}^), we can define a probability measure μ∩ A∈({0,1}^A)⊆^2^A by 

    ∀ B⊆ A   μ∩ A(B)=μ({H⊆| H∩ A=B}).

Then,

    μ_nμ∀ A⊆, |A|<∞   μ_n∩ Aμ∩ A,

where the convergence on the right is the standard one in the Euclidean space ^2^A.

 
The space ({0,1}^ℱ) is compact in the weak* topology.



Since {0,1}^ℱ is compact, the fact can be deduced from the Banach–Alaoglu Theorem (cf. Theorem 3.15 in <cit.>).





 §.§ Pseudo subgroups

For every B⊆ C⊆, the restriction map R_B⊆ C{0,1}^C→{0,1}^B is 
    ∀ A⊆ C   R_B⊆ C(A)=A∩ B.
 When C= we remove it from the notation and write R_B instead of R_B⊆.
For sets K,L⊆ B⊆ we can define

    _B(K,L)={A⊆ B| K⊆ A, A∩ L=∅}.

Note that _(K,L)=(K,L).
 
    Let K,L⊆ B⊆ C⊆. Then R_B⊆ C(_C(K,L))=_B(K,L) and R_B⊆ C^-1(_B(K,L))=_C(K,L).


    Let A⊆ B and A'⊆ C such that A=A'∩ B. As K,L⊆ B, the intersection of A' with K and L depends only on A'∩ B=A. Hence  
    
    A∈_B(K,L)  A'∈_C(K,L).

  Adding this observation to the fact R_B⊆ C is onto proves the desired claim. 


Now, we can pushforward measures along the inverse system of restrictions. Namely, given  B⊆ C⊆ and a probability measure π on {0,1}^C we can define the probability measure R_B⊆ C*π on {0,1}^B by

    ∀ Y ⊆_ Borel{0,1}^B   R_B⊆ C*π(Y)=π(R_B⊆ C^-1(Y)).


 
   Claim <ref> implies that R_B⊆ C is continuous. Therefore,  R_B⊆ C*({0,1}^C)→({0,1}^B) is continuous.





    


This proof is very standard. As oppose to the other basic claims we attach proofs to, this one doesn't seem to contribute to the understanding of the main theorems. So maybe it should be removed. Waiting for your thoughts on it.
 Let f{0,1}^B→ be a continuous function. Define f{0,1}^C→ be the pullback of f along R_B⊆ C, namely 
    
    ∀ A⊆ C   f(A)=f∘ R_B⊆ C(A)=f(A∩ B).

    The pullback is a composition of continuous functions and hence continuous. For every μ∈({0,1}^C),
    
    _H∼μ[f(H)]=_H∼μ[f∘ R_B⊆ C(H)]=_H∼ R_B⊆ C*μ[f(H)].

       So, if  μ_n μ  in ({0,1}^C), then 
       
    _H∼ R_B⊆ C*μ_n[f(H)]=_H∼μ_n[f(H)]_H∼μ[f(H)]=_H∼ R_B⊆ C*μ[f(H)].

       Since this was true for any continuous f, we have that R_B⊆ C*μ_nR_B⊆ C*μ and we are done.






   Let B⊆. A subset  A⊆ B is said to be a pseudo subgroup of B if there is a subgroup H≤ such that R_B(H)=H∩ B=A. Denote by P(B) the collection of pseudo subgroups of B.[Note that this collection is a closed subset of the subsets of B.]

 
    Let A⊆ B⊆. Then, A is a pseudo subgroup of B if and only if ⟨ A⟩∩ B = A, where ⟨ A⟩ is the subgroup of  generated by A. 




    If there is a subgroup H for which  A=R_B(H)=H∩ B⊆ H, then in particular ⟨ A⟩⊆ H. Because A is always contained in ⟨ A⟩∩ B, we can conclude.



    Let A⊆ B⊆ C⊆. If A is not a pseudo subgroup of B, and A'⊆ C restricts to A (i.e., A'∩ B=A), then A' is not a pseudo subgroup of C.


    By Claim <ref>, if A is not a pseudo subgroup of B, then there is a w∈ B such that w∉ A while w∈⟨ A⟩.  Since w∈ B⊆ C, and A'∩ B=A, the same w satisfies that w∉ A' while w∈⟨ A⟩⊆⟨ A'⟩, proving that A' is not a pseudo subgroup of C.





Let B⊆ be a subset.



  * The set 𝒫_B is the collection of all π∈({0,1}^B)  that are concentrated on pseudo subgroups of B, namely π(P(B))=1. Equivalently, π({0,1}^B∖ P(B))=0.

  * The set 𝒬_B is the subset of _B of distributions that are S-invariant when defined. Namely, assuming  K, L ⊆ B are finite subsets such that 
    ∀ s∈ S∪ S^-1    sKs^-1, sLs^-1⊆ B,
 we have π( _B(K,L)) = π(_B( sKs^-1, sLs^-1)).  We call  probability distributions in Q_B pseudo invariant random subgroups over B, or pseudo IRSs in short.

  * Let _B=R_B*^-1(_B) and _B=R_B*^-1(_B).




    Note that _=_=(()) and _=_=().



Let B⊆ C⊆ be subsets.
    
       
        
  * The restriction to B of a distribution over pseudo subgroups of C, is a distribution over pseudo subgroups of B. Namely,
        
    ∀π∈_C   R_B⊆ C*π∈_B.
 
        Similarly, the restriction to B of a pseudo IRS of C is a pseudo IRS of B. Namely, 
        
    ∀π∈_C   R_B⊆ C*π∈_B.

        
  * The (Borel) probability distributions over subgroups of  are the intersection of pull-backs of all distributions over pseudo subgroups in finite subsets. Similarly, the IRSs of  are the intersection of pull-backs of all pseudo IRSs of finite subsets. Namely,
        
    ⋂_B⊆, |B|<∞_B=(()), ⋂_B⊆, |B|<∞_B=().

    




        
      Let π∈_C.  By definition, 
    π({0,1}^C∖ P(C))=0.
 By Corollary <ref>, 
    R_B⊆ C^-1({0,1}^B∖ P(B))⊆{0,1}^C∖ P(C).
 Hence, by the definition of the pushforward, 
      
    R_B⊆ C*π({0,1}^B∖ P(B))   =π(R_B⊆ C^-1({0,1}^B∖ P(B)))
       ≤π({0,1}^C∖ P(C))
       =0,

    and  so R_B⊆ C*π∈_B.  

        Assume further that π∈_C. By Claim <ref>, for every finite K,L⊆ B, 
    R^-1_B⊆ C(_B(K,L))=_C(K,L).
 
        Thus  
        
    R_B⊆ C*π(_B(K,L))   =π(R_B⊆ C^-1(_B(K,L)))
    =π(_C(K,L)).

        If K,L⊆ B such that sKs^-1,sLs^-1⊆ B for every s∈ S∪ S^-1, then the same applies to C since it contains B, and hence
        
    R_B⊆ C*π(_B(K,L))   =π(_C(K,L))
       =π(_C(sKs^-1,sLs^-1))
       =R_B⊆ C*π(_B(sKs^-1,sLs^-1)).

        Therefore R_B⊆ C*π is in _B and (1) is proved.
 

 
  By clause (1), for every B⊆ C⊆ we have _C⊆_B and _C⊆_B. In particular, when C= we have (())⊆_B and ()⊆_B. Hence  
    (())⊆⋂_B⊆, |B|<∞_B,  ()⊆⋂_B⊆, |B|<∞_B.

     We are left to prove the reverse containments. Recall the open cover of the complement of the subgroups described in the proof of Claim <ref>.
     Given π∈({0,1}^) such that π∉(()), we have 
    π((⋃_w,w'∈  C_w,w')∪(⋃_w∈ C_w ) ∪  C_)=π ({0,1}^∖())>0.

    Hence, by the union bound (Boole's inequality), there is either a pair w,w'∈ such that π(C_w,w')>0, or a single element w∈ such that π(C_w)>0, or π(C_)>0. Thus, if B={_,w,w',w· w',w^-1},  then π∉_B. In particular π∉⋂_B, proving the first reverse containment.
     
     Lastly, let π∈⋂_B.  For every finite K,L⊆, there is a larger enough finite set B such that K,L⊆ B and also sKs^-1,sLs^-1⊆ B for every s∈ S∪ S^-1. Since π∈_B,  by Claim <ref> applied to C=, we have for every s∈ S∪ S^-1 that
    
    π((K,L))   =π(R_B^-1(_B(K,L)))
       =R_B*π(_B(K,L))
       =R_B*π(_B(sKs^-1,sLs^-1))
       =π(R_B^-1(_B(sKs^-1,sLs^-1)))
       =π((sKs^-1,sLs^-1)).

    Hence, by Fact <ref>, (2) is deduced.



    

The following claim is never used in the text. It is used as a reference in the introduction, but it seems we do not need it. Should we remove it?

The Aldous–Lyons Conjecture <ref> is true if and only if for every finite subset B⊆ the set  R_B*(_())  is dense in  R_B*(()).


    Let π∈(), and let (B_t)_t=1^∞ be a sequence of subsets of  such that B_t⊆ B_t+1 and ⋃ B_t=. Since  R_B_t*(_())  is dense in  R_B_t*(()) (as subsets of the finite dimensional vector space ℝ^B_t equipped with some norm ‖·‖), we can find π_t∈_() such that ||R_B_t*π-R_B_t*π_t||≤1/t. By Claim <ref>, ({0,1}^) is compact. 
    Moreover, _B is a closed set when B is finite[This is true for any B⊆, but it is straightforward for finite ones, and we only use it in this case.] and R_B⊆ C* is continuous (Corollary <ref>), so by item (2) of Lemma <ref> () is the intersection of closed sets in the weak* topology, 
 and thus it is closed. 
 Hence, π_t has a subsequence with weak* limit π_∞∈() (without loss of generality, the subsequence is π_t itself). For every s≥ t we have 
    
    ||R_B_t*π-R_B_t*π_∞||≤ ||R_B_t*π-R_B_t*π_s||+||R_B_t*π_s-R_B_t*π_∞||.

    Since R_B_t* is continuous, and since π_sπ_∞, there is a large enough s for which 
    
    ||R_B_t*π_s-R_B_t*π_∞||≤1/s.

    We did not choose a specific norm up until now. By the triangle inequality 
    
    ∀σ,σ'∈({0,1}^B_s)   ||R_B_t⊆ B_s*σ-R_B_t⊆ B_s*σ'||_1≤ ||σ-σ'||_1.

    Hence by choosing ||·||=||·||_1, we have 
    
    ||R_B_t*π-R_B_t*π_s||≤ ||R_B_s*π-R_B_s*π_s||≤1/s.

   Therefore for every s≥ t,
    
    ||R_B_t*π-R_B_t*π_∞||≤2/s.

    But the left hand side is independent of s. Thus R_B_t*π=R_B_t*π_∞ for every natural number t. Let K,L⊆ be finite subsets. Since B_t⊆ B_t+1 and ⋃ B_t=, there is a large enough t for which K,L⊆ B_t. By Claim <ref>, 
    
    R_B_t*π(_B_t(K,L))   =π(R_B_t*^-1(_B_t(K,L)))=π((K,L))
    
         R_B_t*π_∞(_B_t(K,L))   =π_∞(R_B_t*^-1(_B_t(K,L)))=π_∞((K,L))

    which implies π((K,L))=π_∞((K,L)).  By Fact <ref>, we deduce π=π_∞. Therefore π_tπ and _() is weak* dense in (), proving Conjecture <ref>. 

    The reverse implication is immediate from the fact the pushforward along restrictions maps R_B*({0,1}^)→({0,1}^B)  are continuous, as was deduced in Corollary <ref>.


 
    Conjecture <ref> is false if and only if there is a finite set K⊆ and a functional φℝ^{0,1}^K→ℝ, such that 
    
    sup{φ(π)|π∈ R_K(_())}<sup{φ(π)|π∈ R_K(())}.



This formulation of Conjecture <ref> should motivate our approach of studying subgroup tests and their values, which are exactly linear functionals of the above format.




The system of finite subsets of   forms a directed system induced by inclusion. The pushforwards along  restriction maps R_B⊆ C* thus form an inverse system of maps between the compact sets (B) for B⊆, |B|<∞. In this language, Lemma <ref> says that 

    (())=_B_B    and    ()=_B _B.





 §.§ Computational aspects of pseudo invariant random subgroups


 
Given A⊆ B⊆ where |B|<∞, there is an algorithm deciding whether A is a pseudo subgroup of B. 





Given a finite set of elements A={ w_1,...,w_k }⊆, and another element w∈, there is an algorithm to decide whether w∈⟨ A⟩. Indeed, one can use the following:


  * Construct Core(A) the core graph of ⟨ A⟩  via the method of Stallings' foldings  <cit.>.

  * Check whether the path that begins at the basepoint of Core(A) and traverses w is closed.

  * If the path is closed, then w∈⟨ A⟩. Otherwise w∉⟨ A⟩.


By Claim <ref>, A is not a pseudo subgroup of B if and only if there exists a w∈ B such that w∉ A but w∈⟨ A ⟩. Hence, a sufficient and necessary condition for A to be a pseudo subgroup of B is to be accepted by the following algorithm:

    
  * Construct  Core(A).
    
  * For every w∈ B satisfying w∉ A, reject if w is a closed path in Core(A).
    
  * Accept if A passed (2) for every w∈ B.




    Recall that a free group is LERF (locally extended residually finite), i.e., for every finite set A={w_1,...,w_k} and every w∈ such that w∉⟨ A⟩, there exists a finite index subgroup H for which 

    A⊆ H and w∉ H.

This was originally proved by M. Hall <cit.>, and can be proven using the method of Stallings' foldings <cit.>. It follows that for  every A a pseudo subgroup of a finite B, there exists a finite index subgroup H for which R_B(H)=A.



    Let B⊆ be a finite set. Then, the set _B of distributions over pseudo subgroups of B and the set _B of pseudo IRSs over B are computable polytopes in ℝ^{0,1}^B. Namely, they are defined by (computable) integral linear equations and inequalities.


         Recall that _B⊆_B⊆({0,1}^B)⊆^{0,1}^B. Namely, all these distributions  are vectors which are indexed by subsets of B.     
     We describe the defining system of equations and inequalities forcing a vector π∈ℝ^{0,1}^B to belong to _B. First, it needs to be a probability measure, hence  
        
    ∀ A⊆ B   π(A)   ≥ 0,
    ∑_A⊆ Bπ(A)   =1.

        Further, for every A⊆ B, we can run the algorithm in Claim <ref> to decide whether A is a pseudo subgroup, and if not to add the constraint π(A)=0. This is the defining system of _B. To get the defining system of _B, we need to add the following:
        For every K,L⊆ B, we can check whether for all s∈ S∪ S^-1, the sets sKs^-1 and sLs^-1 are also contained in B, and if so add the constraints
        
    ∀ s∈ S∪ S^-1   ∑_A∈_B(K,L)π(A)_π(_B(K,L))=∑_A∈_B(sKs^-1,sLs^-1)π(A)_π(_B(sKs^-1,sLs^-1)).

        








 








§ MAIN THEOREM I: APPROXIMATING THE VALUES OF SUBGROUP TESTS
 

Recall the definitions of challenges, tests and values from Section <ref>.
Let  be a test over the generating set S, with challenges (K_i;D_i)_i∈ Q and rational distribution μ over Q. Let K=⋃_i∈ Q K_i, which is a finite subset of (S)=. We can extend the definition of (,·) in the following way: For every  B satisfying K⊆ B⊆, and for every distribution π∈({0,1}^B),  let 

    (,π)=_A∼π_i∼μ[D_i(A∩ K_i)].

The next claim is formalizing the following intuitive idea: Since D_i depends only on the restriction of H to K_i, then replacing π by its push-forward along any restriction R_B will not change its value against .


Under the extended definition of the value, for every B and C such that K⊆ B⊆ C⊆, and for every π∈({0,1}^C), we have

    (,π)=(,R_B⊆ C*π).




    By definition, sampling A∼ R_B⊆ C*π is the same as sampling A'∼π and outputting A=A'∩ B.
    Also, K_i=B∩ K_i since K_i⊆ K⊆ B. Hence,
    
    (,R_B⊆ C*π)   =_A∼ R_B⊆ C*π_i∼μ[D_i(A∩ K_i)]
       =_A'∼π_i∼μ[D_i((A'∩ B)∩ K_i)]
       =_A'∼π_i∼μ[D_i(A'∩ K_i)]
       =(,π).



Recall the first clause of Theorem <ref>: There is a Turing machine M_1 that takes as an input the encoding of a test , and outputs an infinite sequence of non-decreasing numbers (α_t)_t=1^∞⊆ [0,1]  such that lim_t→∞α_t=_().




 
Let σ S→(X) be a finite action. The value of Φ(σ) against  is 
 
    (,Φ(σ))   =_H∼Φ(σ)_i∼μD_i(H∩ K_i)
       =∑_x∈ X∑_i∈ Qμ(i)/|X|· D_i((σ,x)∩ K_i)

 which is a rational combination of  evaluations of D_i. Since the sets K_i, functions D_i and coefficients μ(i) can be calculated in finite time from  the encoding of , and (σ,x)∩ K_i can be calculated from σ for every x∈ X, the value of Φ(σ) against  can be computed in finite time. 

  There is a computable countable enumeration of all (equivalence classes of) finite (S)-actions. To see that, note that a map σ S→(n) is defined by the tuple (n ; {σ(s)}_s∈ S). Since the elements of (n) can be ordered (e.g., lexicographically), as well as the elements of S, the infinite set of tuples (n ; {σ(s)}_s∈ S) inherits a countable enumeration. Denote by σ_t the t^ th action according to this enumeration.
 For t∈ℕ, let 
 
    α_t=max_s≤ t{(,Φ(σ_s))}.

 By our discussion so far, the number α_t can be computed in finite time. Moreover, (α_t)_t=1^∞ is a non-decreasing sequence by definition. 
 Since  Im(Φ)=_(), the sequence (α_t)_t=1^∞ tends to the supremum of (,·) over _(), which is, by (<ref>), _().


We now recall the second clause of Theorem <ref>:  There is a Turing machine M_2 that takes as an input the encoding of a test , and outputs an infinite sequence of non-increasing numbers (β_t)_t=1^∞⊆ [0,1]  such that lim_t→∞β_t=_().







 Let K=⋃ K_i as before, and let B⊆ be a finite subset containing K.  Recall the definition of _B the pseudo invariant random subgroups over B (Definition <ref>), and its pre-image _B. Recall also  Claim <ref> and the discussion before it, where we extended the notion of the value to probability distributions over  subsets of  that contain K. The B-approximation to the ergodic value of  is defined to be

    _B() = sup{(,π)|π∈_B}=sup{(,π)|π∈_B}.


Let {B_t}_t=1^∞ be a sequence of finite subsets of , all containing K, such that B_t⊆ B_t+1 and ⋃_t=1^∞ B_t=. 
We now claim the following:

    
  * _B_t() is computable.
    
  * The sequence _B_t() is non-increasing.
    
  * lim_t→∞_B_t()=_().


By Lemma <ref>, _B is defined by a computable integral system of equations and inequalities. Thus, maximizing a linear functional (,·) over it can be done using linear programming in finite time, proving (1).

By Lemma <ref>, whenever B⊆ C⊆ we have R_B⊆ C*(_C)⊆_B. Hence _B_t+1⊆_B_t, which implies _B_t+1()≤_B_t(), proving (2).

Moreover,  ()=_⊆_B_t  which implies _B_t()≥_().
On the other hand, by Claim <ref> the space ({0,1}^) is compact, and since _B are closed sets they are also compact. Hence there is a distribution π_t∈_B_t satisfying _B_t()=(,π_t). By the compactness of ({0,1}^), there is a weak limit to some subsequence of (π_t)_t=1^∞, which we denote by π_∞. Without loss of generality the subsequence is the original sequence. Since π_s∈_B_t for every s≥ t, and since   _B are closed, we deduce that π_∞∈⋂_B_t=(). By the definition of a weak limit, since (,·) is a continuous functional, we have 

    lim_t→∞_B_t()=lim_t→∞(,π_t)=(,lim_w^*π_t)=(,π_∞)≤_(),

proving (3). Therefore, choosing β_t=_B_t() finishes the proof.




§ EXAMPLES OF TESTS
 




 §.§ Relations verification test
  Given a finite subset R⊆(S), the R-verification test 𝒱_R checks whether R is contained in the sampled subgroup H. It contains a single challenge (R;D) with D= 1_R, namely

    ∀ A⊆ R   D(A)=
        1     A=R,
    
        0     A≠ R.


The sofic value of _R is always 1, e.g., using the Dirac measure concentrated on the group  itself (which associates via Φ with the trivial action of  on a singleton). Moreover, if the conjugation orbit of a finite index subgroup H passes _R with probability 1, then R⊆⋂_w∈ wHw^-1. 




Though the value of the R-verification test does not give us much information about R, we will see  in Section <ref> that the potential robustness of this test is an important property of R.
Moreover, this test has variations where you sample r∈ R according to some distribution and check only that r∈ H and not that all of R is in H. See for example CL_part1,CVY_efficient for more on that.




 §.§ Separation test
  
Given two finite subsets R,L⊆(S), the (R,L)-separation test 𝒱_R||L checks whether R is contained in the sampled subgroup H but L∖ R is not. It contains a single challenge (R∪ L;D) with D= 1_R, namely

    ∀ A⊆ R∪ L   D(A)=
        1     A=R,
    
        0     A≠ R.


 
A group Γ is residually finite if for every finite subset L⊆Γ where _Γ∉ L, there is a finite index normal subgroup N⊴Γ such that L∩ N=∅.




    A finitely presented group Γ=⟨ S|R⟩ is residually finite if and only if for every finite L⊆ such that L∩⟨⟨ R⟩⟩=∅[The subgroup ⟨⟨ R⟩⟩ is the smallest normal subgroup containing R.] there is a finitely described IRS passing _R||L with probability 1. 



    Let Γ=(S)/⟨⟨ R⟩⟩ be residually finite, and L⊆ a finite set disjoint from ⟨⟨ R⟩⟩. By Definition <ref>, there is a finite index normal subgroup N⊴Γ such that L⟨⟨ R⟩⟩∩ N=∅. Using the sequence of epimorphisms
    
    ↠Γ↠Γ/N

    we can define the left action of  on X=Γ/N which passes _R||L with probability 1. 
    
    On the other hand, given a finite subset _Γ∉ L⊆Γ, let  L'⊆ be any set of representatives for L. Since _Γ∉ L, we have L'∩⟨⟨ R⟩⟩=∅. 
    Hence, by our assumption,  there is an  action ρ S→(X) with (_R||L',Φ(ρ))=1. Without loss of generality, ρ is transitive (otherwise, its restriction to any orbit will do).  
    The fact (_R||L',Φ(ρ))=1 implies in particular that R⊆(ρ,x) for any x∈ X. Thus, ρ factors through Γ and defines an action ρ'Γ→(X) by ρ'(w⟨⟨ R⟩⟩)=ρ(w). By the fact (_R||L',Φ(ρ))=1, the permutation ρ(w) has no fixed points for every w∈ L', and thus ρ'(u) has no fixed points for every u∈ L. Hence N=⋂_x∈ X(ρ',x) is a normal subgroup of Γ satisfying N∩ L=∅, as required to deduce Γ is residually finite.



    Let X be a finite set. Given two permutations σ,τ∈(X), we define their normalized Hamming distance to be

    d_H(σ,τ):=|{ x∈ X|σ(x)≠τ(x)}|/|X|=ℙ_x∈ X[σ(x)≠τ(x)].





    A finitely presented group Γ=⟨ S|R⟩ is sofic if for every L⊆(S) such that L∩⟨⟨ R⟩⟩=∅, and for every >0, there is a finite set X and an action ρ S→(X) such that 
    
    ∀ r   ∈ R   d_H(ρ(r),_X)≤,
    ∀ w   ∈ L   d_H(ρ(w),_X)≥ 1-.




It is straightforward to see that soficity is a relaxation of residual finiteness. 



    A finitely presented group Γ=⟨ S|R⟩ is sofic if and only if for every finite L⊆ such that L∩⟨⟨ R⟩⟩=∅ we have _(_R||L)=1.



Assume Γ is sofic, and let  L⊆ be a finite set such that L∩⟨⟨ R⟩⟩=∅. For every >0, there is an action ρ_ S→(X_), such that 
  
    ∀ r   ∈ R   d_H(ρ_(r),_X_)≤,
    ∀ w   ∈ L   d_H(ρ_(w),_X_)≥ 1-.

    Hence,
    
    (_R||L,Φ(ρ_))   =_x∈ X_[∀ r∈ R,∀ w∈ L  ρ_(r).x=x, ρ_(w).x≠ x]
       ≥ 1-∑_r∈ R_x∈ X_[ρ_(r).x≠ x]-∑_w∈ L_x∈ X_[ρ_(w).x=x]
       ≥ 1-∑_r∈ Rd_H(ρ_(r),_X_)-∑_w∈ L(1-d_H(ρ_(w),_X_))
       ≥ 1-|R|-|L|.

    Since |R| and |L| are fixed, and >0 is arbitrarily small, we have _(_R||L)=1.

    On the other hand, if for every L we have _(_R||L)=1, then for every >0 there is an action ρ S→(X) with (_R||L,Φ(ρ))≥ 1-. Now, since 
    
    (_R||L,Φ(ρ))   =_x∈ X[∀ r∈ R,∀ w∈ L  ρ(r)x=x, ρ(w)x≠ x]
       ≤min(min_r∈ R(_x∈ X[ρ(r)x=x]), min_w∈ L(_x∈ X[ρ(w)x≠ x]))
       =min(min_r∈ R(1-d_H(ρ(r),_X)), min_w∈ L(d_H(ρ(w),_X))),

    we deduce that
 
    ∀ r   ∈ R   d_H(ρ(r),_X)≤,
    ∀ w   ∈ L   d_H(ρ(w),_X)≥ 1-.

    and Γ is sofic.



 §.§ Formula satisfaction test

Let φ(y_1,...,y_n) be a boolean formula. The test _φ is defined over the formal set of generators S={x_1,...,x_n}. It has a single challenge (K;D), where K=S. Note that the characteristic function of every subset A⊆ S is already in the form 1_A{x_1,...,x_n}→{0,1}, and thus induces an assignment for the formal variables. The decision predicate D accepts A⊆ S if 1_A induces a satisfying assignment for φ, i.e.,

    D(A)=φ( 1_A(x_1),..., 1_A(x_n)).

It is straightforward that (_φ)=1 if and only if φ is satisfiable. 

There is a more interesting version of this test, when the formula φ is a 3CNF. Recall that a 3CNF is an and of clauses of the form φ_t=y_i^_1 y_j^_2 y_k^_3, where y_i,y_j,y_k are taken from a fixed set of formal generators (as before), _1,_2,_3∈{0,1}, and we interpret y^0=y and y^1=¬ y. Now, checking that φ is satisfied by 1_A is the same as going over all the clauses and checking that they are all satisfied. Thus, we can define a randomized version of this check by associating a challenge (K_t;D_t) with each clause φ_t, by letting K_t={x_i,x_j,x_k} and D_t(A)=φ_t( 1_A(x_i), 1_A(x_j), 1_A(x_k)). Then, by choosing a uniform distribution over challenges,  the associated test _φ checks whether a uniformly chosen clause is satisfied by 1_A. The famous PCP theorem <cit.> says that any 3CNF φ can be (efficiently) transformed into a not much larger 3CNF φ', such that if _(_φ)=1, then _(_φ')=1, and if _(_φ)=0, then  _(_φ')≤1/2.
The scaled up version of this argument shows that any  problem (see <cit.>) can be reduced to approximating the sofic value of a test.[This last remark is meaningless as long as we do not specify the exact way we encode tests. The point  is that there is a natural way to encode tests such that  can be reduced to approximating the sofic value immediately from <cit.>. In any case,  this paper is devoted to proving a much stronger conclusion, which is that approximating the sofic value is as hard as the Halting Problem.]




    





§ ROBUSTNESS
 
 In this section we define  a rigidity property of tests called robustness (see Definition <ref>). Loosely, a test is robust if every almost optimal strategy against it is close to an optimal one. Though this notion is natural, one needs to clarify what is the measure of distance between strategies. 



 §.§ Edit distance

We first define a generalized version of the normalized Hamming distance (<ref>). Let X and Y be two finite sets, and assume X⊆ Y. Given two permutations σ∈(X) and τ∈(Y), we define their normalized Hamming distance with errors to be

    d_H(σ,τ)=1-|{ x∈ X|σ(x)=τ(x)}|/|Y|.

If for every y∉ X we let σ(y)=error ∉ Y,
then we can define equivalently 

    d_H(σ,τ)=ℙ_y∈ Y[σ(y)≠τ(y)],

which is the way we defined the normalized Hamming distance in (<ref>). Since the generalized version is the same as the usual Hamming distance when Y=X, we use the same notation for both and just call them the Hamming distance from now on.

Let S→ℝ_>0 be a function, which we call the significance function. 
Given two actions ρ S→(X) and φ S→(Y), we  define their -weighted distance as follows:

    d^
    (ρ,φ)=∑_s∈ S(s)· d_H(ρ(s),φ(s)).[Though the Hamming distance is always bounded by 1, the weighted distance may be larger than 1.]

For every θ∈(Y), let φ^θ S→(Y) be φ^θ(s)=θφ(s) θ^-1.
Now, the -weighted edit distance[
There is a more graph theoretical way of viewing the edit distance, in the spirit of the  section <ref>. It essentially measures how much one Schreier graph needs to be changed to get to the other graph. For more on that, see Section 4 of <cit.> and the Γ-graph notion in <cit.>.] of ρ and φ is  the following minima: 

    d^_ edit(ρ,φ)=min_θ∈(Y)(d^(ρ,φ^θ)).

Note that the edit distance is independent of the specific embedding of X in Y.
The edit distance induces a metric on _(). For every π,π' ∈_(), let

    d^_ edit(π,π')=inf{ d^_ edit(ρ,φ) |ρ∈Φ^-1(π),φ∈Φ^-1(π')} ,

where recall the definition of Φ in (<ref>).






  §.§.§ A graph theoretic perspective
 


 Let Γ be a  group with S a finite  generating set. Let X be a set and σΓ→(X) an action of Γ on X. The generalized Schreier graph  (σ,S) is a directed graph with vertex set X and edges labeled by S. The edges of (σ,S) are defined as follows: For every x∈ X  and s∈ S, there is a directed edge from  σ(s).x to x labeled by s, namely

    σ(s).x x.

We may say that an edge is labeled by some inverse of s∈ S, such as xy. By that we mean the edge labeled by s and oriented in the other direction, namely yx. This is natural as σ(s^-1)=σ(s)^-1. Moreover, when S generates Γ, it includes words with inverses, and so when one associates words in the generators to paths in a labeled graph, this interpretation is natural (cf. <cit.>). 


    This  generalizes the usual notion of a Schreier coset graph (Γ,H,S) by choosing σΓ→_H\^Γ to be the natural action of Γ on right H-cosets. 
 
The Schreier graph does not provide any new data about σ, but it  provides a graph theoretic perspective for approaching IRSs.
For example, a fact that we repeatedly use is that orbits of the action are in correspondence with the connected components of the Schreier graph. Moreover, the edit distance is better understood as a distance between graphs: Given two labeled graphs, one wants to transform one to the other. Whenever the endpoint of an edge labeled by s is changed, there is an associated cost (s). The same cost is incurred when deleting completely or adding such an s-labeled edge. The goal is to minimize the cost of moving between the graphs (up  to isomorphism). The edit distance is exactly the minimal cost of such a transformation.







  §.§.§ Topology induced by the edit distance

 It turns out that regardless of the significance function , the topology induced by the edit distance is strictly stronger than the weak^* topology:

  We have the following:


  * Given two sequences π_n,π'_n∈_() such that π_n converges  to π_∞∈() in the weak^* topology and lim_n→∞d^_ edit(π_n,π'_n)=0, then π'_n converge to the same π_∞ in the weak^* topology.

  * There are sequences π_n∈_() that converge in the weak^* topology but are not Cauchy sequences in the edit distance topology.




    The above proposition should be contrasted with the amenable case, studied in <cit.>, where the authors showed that the converse of (1) is true when the free group is replaced by an amenable group. See <cit.> for a different proof of the above clause (2).





Let min() be the minimum value of . Since S is finite, min()>0. So d^_ edit≥min()d_ edit where d_ edit is the edit distance with significance function equal to the constant 1. So lim_n→∞d_ edit(π_n,π'_n)=0. This reduces the problem to the ordinary edit distance.

Let K,L be finite subsets of . By Remark <ref>, we have

    lim_n→∞π_n((K,L)) = π_∞((K,L)).

So it suffices to prove

    lim_n→∞ |π_n((K,L))-π'_n((K,L))| = 0.

Let r>0 be a large enough radius so that K∪ L is contained in the ball B(r) ⊂ of radius r with respect to the word metric induced by S.

Because π_n and π'_n are finitely described, there are actions ρ_n S→(X_n) and ρ'_n S→(Y_n) which induce π_n and π'_n respectively. We may assume that for each n, either X_n ⊂ Y_n or Y_n ⊂ X_n. After conjugating if necessary, we may further assume

    d_ edit(ρ_n,ρ'_n) = d(ρ_n,ρ'_n) = ∑_s∈ S d_H(ρ_n(s),ρ'_n(s)).


Given an integer k ≥ 1, let (k) be the set of all v∈ X_n ∪ Y_n such that there exists g in the ball B(k)⊂ with ρ_n(g)^-1v ρ'_n(g)^-1v. The definition of edit distance gives

    |(1)| ≤ d_ edit(ρ_n,ρ'_n) |X_n ∪ Y_n|.

Suppose v ∈(k) ∖(k-1) for some k≥ 2. Then there is a g ∈ B(k) such that ρ_n(g)^-1v ρ'_n(g)^-1v. Let g=s_1⋯ s_k with s_i ∈ S ∪ S^-1. Since v ∉(k-1), if h=s_1⋯ s_k-1, then w:=ρ_n(h)^-1v = ρ'_n(h)^-1v and ρ_n(s_k)^-1w ρ'_n(s_k)^-1w. Hence w ∈(1) and so v ∈ B(k-1)(1). Therefore,

    |(k)| ≤ |B(k-1)| |(1)| ≤ |B(k-1)|d_ edit(ρ_n,ρ'_n) |X_n ∪ Y_n|.

Since |B(k-1)| is independent of n, and d_ edit(ρ_n,ρ'_n) 0, we can deduce that

    lim_n→∞|(k)|/|X_n ∪ Y_n| = 0

for all k. 

Because K ∪ L ⊂ B(r), if v ∉(r), then (ρ_n, v) ∈(K,L) if and only if (ρ'_n, v) ∈(K,L). Hence

    |π_n((K,L))-π'_n((K,L))| ≤|(k)|/|X_n ∪ Y_n|→ 0

as n→∞. This finishes the proof of (1).

To prove item (2), recall that if G=(V,E) is a graph then a subset I ⊂ V is independent if no edge in E has both endpoints in I. If G is finite then the independence ratio of G is α(G)=|I|/|V| where I is an independent set of maximum cardinality. If ρ:S →(n) is an action then we let α(ρ)=α(G) where G is the associated action graph.

For any actions ρ and ρ',  we have d_edit(ρ,ρ') ≥ |α(ρ)-α(ρ')|. So it suffices to show there exist action sequences {ρ_n}_n, {ρ'_n}_n whose corresponding IRSs weak^* converge to the same IRS and  |α(ρ)-α(ρ')|>c for some constant c>0.

By using a first moment argument, it can be shown that if ρ_n:S →(n) is chosen uniformly at random and if G_n is the associated action graph then with high probability α(G_n)<1/2 - c where c>0 is a constant depending only on |S| (which we assume is at least 2). Moreover,  there are sets Ω_n ⊂(n)^S with |Ω_n|/|(n)^S| → 1 as n→∞ such that if ρ_n ∈Ω_n for all n and π_n are the corresponding IRSs then π_n weak^* converges to the trivial subgroup {}. This can be derived by estimating the expected number of short cycles in the graph G_n <cit.>. A more precise estimate on Ω_n is obtained in <cit.>.

Therefore, there exists a sequence {ρ_n} of actions such that if π_n are the corresponding IRSs then π_n weak^* converges to the trivial subgroup {} and α(G_n) < 1/2 - c.

On the other hand, there exist ρ'_n:S →(n) whose action graphs are bi-partite such that if π'_n is the corresponding IRS then π'_n also weak^* converges to the trivial subgroup {}. Since the independence ratio in this case is 1/2, it follows that the sequence π_1,π'_1,π_2,π'_2,… cannot be Cauchy in the edit distance.

















 §.§ Robustness

Given s∈ S, we can define a map v_s→ℕ that given a reduced word w∈ counts the number of appearances of s or s^-1 in w. Namely, if w=∏_j=1^ℓ s_j^_j,  where s_j∈ S and _j∈{± 1}, is a reduced word, then v_s(w)=∑_j=1^ℓ 1_s(s_j).

Let  be 
a test with its usual associated data S, (K_i;D_i)_i∈ Q and μ. We define _ S→ℝ_>0, the significance function associated with , as follows: 

    _(s)=_i∼μ[∑_w∈ K_i v_s(w)].


 
    Let >0, and  be 
a test. Let X and Y be finite sets, and ρ S→(X) and φ S→(Y) be actions. Assume d^__ edit(ρ,φ)≤. Then |(,Φ(ρ))-(,Φ(φ))|≤.



    Assume without loss of generality that X⊆ Y and that d^__ edit(ρ,φ)=d^_(ρ,φ). Let i∈ Q. For every w=∏_j=1^ℓ s_j^_j∈, we have
    
    _y∈ Y[ 1_(ρ,y)(w)≠ 1_(φ,y)(w)]≤_y∈ Y[ρ(w).y≠φ(w).y].

Inequality (<ref>) is a consequence of the following implication: If  ρ(w) acts on y the same way as φ(w), then in particular  w is either in both (ρ,y) and (φ,y) or in none of them. For 1≤ k≤ℓ, let w_k be the suffix of w from position k, namely w_k=∏_j=k^ℓ s_j^_j. Moreover, let w_ℓ+1 be the empty word _. Then 

    _y∈ Y[ρ(w).y≠φ(w).y]≤_y∈ Y[⋁_1≤ k≤ℓρ(w_k). y ≠φ(w_k). y].

Inequality (<ref>) is a consequence of  the following implication: If the endpoint of the path beginning at y and labeled by w^-1 in (ρ,S) is different than the endpoint of  the same path in (φ,S), then the two paths diverged at some point. Now,
   
    _y∈ Y[⋁_1≤ k≤ℓρ(w_k). y ≠φ(w_k) .y]≤∑_k=1^ℓ_y∈ Y[ρ(w_k). y ≠φ(w_k). y ρ(w_k+1). y = φ(w_k+1). y].

Inequality (<ref>) is a consequence of the union bound and the logic tautology ⋁_j=1^m φ_j=⋁_j=1^m(φ_j¬φ_j-1), where φ_0= False. But, since 
   
    ρ(w_k) .y ≠φ(w_k) .y ρ(w_k+1) .y = φ(w_k+1) .y ρ(s_k^_k)∘ρ(w_k+1).y≠φ(s_k^_k) ∘ρ(w_k+1).y,

   and since ρ(w_k+1).y is uniformly distributed given that y is uniformly distributed, we can deduce that 
    
    _y∈ Y[ρ(w_k) .y ≠φ(w_k) .y ρ(w_k+1). y = φ(w_k+1) .y]
        ≤_y∈ Y[ρ(s_k) .y≠φ(s_k) .y ]_=d_H(ρ(s_k),φ(s_k)).

   All in all, 

    _y∈ Y[ 1_(ρ,y)(w)≠ 1_(φ,y)(w)]≤∑_j=1^ℓ d_H(ρ(s_j),φ(s_j))=∑_s∈ Sv_s(w)d_H(ρ(s),φ(s))

Hence, by the union bound and inequality (<ref>), we have

    _i∼μ_y∈ Y[(ρ,y)∩K_i≠(φ,y)∩K_i]   ≤_i∼μ[∑_w∈ K_i∑_s∈ S v_s(w)d_H(ρ(s),φ(s))]
       =∑_s∈ S_i∼μ[∑_w∈ K_iv_s(w)]_(<ref>)d_H(ρ(s),φ(s))
       =∑_s∈ S_(s)d_H(ρ(s),φ(s))
       =d^_(ρ,φ)≤.

Lastly, by combining  

    |(,ρ)-(,φ)|   ≤_i∼μ_y∈ Y[D_i((ρ,y)∩K_i)≠ D_i((φ,y)∩K_i)]
       ≤_i∼μ_y∈ Y[(ρ,y)∩K_i≠(φ,y)∩K_i]

with 
 (<ref>), we deduce the claim.



    For those familiar with  robustness of non-local games, d^__ edit plays a somewhat analogous role to the state-dependent distance (cf. Section 4 of <cit.>, for example).


Robustness is a reverse implication  to the one in Claim <ref>.


Let   be a test and δℝ_≥0→ℝ_≥0 a function satisfying δ()0. We say that  is δ-robust if: 


  * There exists an optimal strategy, namely π∈_() such that (,π)=_().

  * Almost optimal strategies are close to optimal ones, namely, if  π'∈_() satisfies (,π')≥_()-, then there is an optimal strategy π such that d^__ edit(π,π')≤δ().




Let Γ=⟨ S|R⟩ be a finitely presented group. Then, Γ is said to be δ-homomorphism stable, where δ()0, if for every ρ S→(X) where  _r∈ R[d_H(ρ(r),_X)]≤ there is a Γ-action φ S→(Y), where X⊆ Y and such that _s∈ S[d_H(ρ(s),φ(s))]≤δ().


    This notion is more commonly known as flexible pointwise group stability in permutations. See GlebskyRivera,ArzhantsevaPaunescu,BLT,BeckerLubotzky,CL_part1,CVY_efficient for various results in this theory.


    The group Γ=⟨ S|R⟩ is δ-homomorphism stable if and only if the  R-verification test _R is δ'-robust. The exact relation between δ and δ' can be calculated explicitly — they are constant multiples of one another, where the constants depend only on S and R.





§ TAILORED NON-LOCAL GAMES


We commonly use both {0,1} and _2 for the set with 2-modular arithmetic.



 §.§ Non-local games preliminaries

We repeat the definitions which appeared in Section <ref> of the introduction. Since this paper is not focused on non-local games, we treat them somewhat technically. For formal definitions associated with the complexity class ^*, we refer to <cit.>. For the connection between ^* and nonlocal games, a good starting point is <cit.>.


   A (synchronous) non-local game  consists of a finite graph G=(V,E), a length function ℓ V→ℕ, formal sets of generators S_x={^x,i| 1≤ i≤ℓ(x)} for every vertex x∈ V, a distribution μ over the edges E, and decision functions D_xy{0,1}^S_xy→{0,1} for every edge xy∈ E, where S_xy=S_x∪ S_y. We denote by S the set ⋃_x∈ V S_x.


   Anyone familiar with the definition of a non-local game will immediately notice that the standard formalism is different than the one we chose here. Usually, a (synchronous) non-local game is defined as a pair of finite sets X,A, commonly referred to as the question and answer sets respectively, together with a   distribution μ over pairs of questions X× X and a decision predicate D X× X× A× A→{0,1} satisfying D(x,x,a,b)=0 for any x∈ X and a≠ b∈ A. 

   It is quite straightforward to move between the definitions.   A way one can extract the data of Definition <ref> from the above is as follows: Let ℓ be the constant function Λ=⌈log |A|⌉, and fix an embedding of A into {0,1}^Λ. The vertices V of the underlying graph G will be X, and the support of μ will be the edge set E⊆ X× X. The formal generator ^x,i corresponds to the i^ th bit of the answer a when x∈ X is asked as a question. 
   Then, given that x,y were asked, a pair of answers a,b can be encoded as a map γ S_x∪ S_y→{0,1}, where 
   
    γ|_S_x=a   and  γ|_S_y=b .
 
  The  notation (<ref>) will be used repeatedly in the text. 
   Lastly, D_xy(γ)=D(x,y,a,b), where γ is the aforementioned encoding. 

   In the other direction, we can define A to be all bit strings shorter than Λ=max_x∈ Vℓ(x), X to be the vertex set V, and μ stays the same. Finally, D(x,y,a,b)=D_xy(γ) when a,b are of the correct lengths, i.e. a=ℓ(x),b=ℓ(y), and is 0 when either of them is of the incorrect length. Since we consider only synchronous strategies, the condition D(x,x,a,b) is never checked in practice for a≠ b and we can assume it was satisfied beforehand.
Because of this correspondence, we may refer to ℓ as the answer length function.



Non-local games are often dramatized as two-prover interactive proofs with one round <cit.>. This perspective  may help some readers to absorb the upcoming technicalities better. 
Two players, that can share some resources beforehand (random bits in the classical case and entangled qubits in the quantum case), are separated spatially — e.g., they are seated in far away rooms. 
A referee  samples a pair of questions and sends one to each player. 
The players then use their shared resources to come up with answers, and send them back to the referee. 
The referee then decides, using the decision predicate, whether the players won or lost. 
The decision predicate as well as the distribution over possible questions are assumed to be known to the players beforehand.  
We demonstrate this dramatization in Example <ref>.
   


The following is the quantum mechanics' analogue of a probability distribution. Similar to the way probability distributions (over finite sets) are collections of non-negative real numbers that add up to 1, the quantum analogue would be “non-negative” matrices that add up to the identity. 


    A positive operator valued measure (POVM) of dimension n with outcomes in a set A  is a mapping A→ M_n× n(ℂ) such that _a is a positive (semi-definite) matrix for every a∈ A and ∑_a∈ A_a=_n. It is called projective, or a projection valued measure (PVM), if in addition _a is an orthogonal projection for every a∈ A, namely (_a)^2=_a=(_a)^*, where * is the conjugate transpose operation. 
Given that  is a PVM and A={0,1}^Λ,  we define for 1≤ i≤Λ the i^ th marginal of  to be ^i{0,1}→ M_n× n(ℂ)  by 

    ^i_0=∑_aΛ→{0,1}
     a_i=0_a   and  ^i_1=∑_aΛ→{0,1}
     a_i=1_a.

Thus, _a=∏_i=1^Λ^i_a_i. Furthermore, the matrix ^i=^i_0-^i_1 is an order 2  unitary, which we call the i^ th binary observable. All in all, a PVM can be given either as a map {0,1}^Λ→ M_n× n(ℕ), or as a map [Λ]→ U(n) such that the images of  are commuting involutions — i.e., they square to the identity.

As its name suggests, every POVM  defines a probability distribution over its answer set A as follows

    [a  is sampled]=τ(_a),

where τ is the dimension normalized trace on n× n matrices. Such an answer is said to be sampled according to  and we denote it by a∼.


    In the case of a PVM with outcome set A={0,1}^Λ, the observables {^i}_i=1^Λ generate a unitary representation of _2^Λ. The _a's in this case can be read from the unitary representation using the Fourier transform of the representation. Hence, for the case of A={0,1}^Λ, a PVM can be given as a collection of mutually perpendicular ortohgonal projections that sum up to the identity, or as a unitary representation of _2^Λ, and one can move from one perspective to the other without losing information.


     A (synchronous quantum) n-dimensional strategy for  is a map ρ S→ U(n), where the images are involutions, and ρ(S_x) commutes for every fixed x∈ V — by saying that a set of matrices (or later permutations) commutes, we mean that every pair of elements in this set commute.
     Another way of viewing strategies is by saying that they associate with every vertex x∈ V an n-dimensional PVM (in observable form).
     The strategy ρ is said to be commuting along edges if ρ(S_xy) commutes for every edge xy∈ E. 

     
     Such a strategy defines for every edge xy∈ E a probability distribution over functions γ S_xy→{0,1} as follows:
     
    [γ S_xy→{0,1}  is sampled]=τ(∏_∈ S_x^_γ()∏_∈ S_y^_γ()),

where τ is again the dimension normalized trace on n× n matrices, ^_0ℂ^n→ℂ^n is the projection on the (+1)-eigenspace of ρ(), and ^_1ℂ^n→ℂ^n is the projection on its (-1)-eigenspace.[Note that ρ()=^_0-^_1, which exactly shows how to move from this observable form of the PVMs induced by ρ back to its projection based form.] We say that γ was sampled according to ρ if it has this distribution, and we denote it by γ∼ρ.

For γ S_xy→{0,1}, let ^x_γ=∏_∈ S_x^_γ(). Then, the map ^xy{0,1}^S_xy→ M_n× n(ℂ) defined by ^xy_γ=^x_γ^y_γ^x_γ  is  a POVM. Thus, γ∼ρ is a special case of sampling according to a POVM (<ref>), which in this case is ^xy. In the spirit of the notation in (<ref>), it may be appropriate to write ^x_γ=^x_a and ^y_γ=^y_b.


    The images of  according to ρ are usually called binary observables, and the value γ() is usually called the measurement outcome.


 
    There is a procedural (i.e., algorithmic) way of  sampling  γ S_xy→{0,1} according to ρ: 
Since ρ(S_x)⊆ U(n) commutes, these matrices have a mutual orthonormal basis of eigenvectors B_x. Similarly, ρ(S_y) have a mutual orthonormal basis of eigenvectors B_y.
 First, sample v⃗∈ B_x uniformly at random. 
 Then, sample u⃗∈ B_y according to the  squared length of its projection on v, namely 
    [u⃗∈ B_y  is sampled|v⃗∈ B_x  was sampled]=|⟨v⃗|u⃗⟩ |^2,

        where ⟨·| ·⟩ is the standard inner product on ℂ^n, i.e., ⟨v⃗|u⃗⟩=∑_i=1^n v_i· u_i.
 Finally,  γ S_xy→{0,1} is determined as follows — for ∈ S_x, γ()=
        0    ρ()v⃗=v⃗,

        1    ρ ()v⃗ =-v⃗,, and for ∈ S_y, γ()=
        0    ρ()u⃗=u⃗,

        1    ρ ()u⃗ =-u⃗..
        All in all, 
        
    [ (v⃗,u⃗)  is sampled]=|⟨v⃗|u⃗⟩|^2/n,

        which is independent of the ordering between x and y. Note also that the probability a specific γ is measured is  independent of the  chosen bases B_x and B_y.



The function γ S_xy→_2 is sampled in the above procedure if for the resulting pair (v⃗,u⃗) we have v⃗∈( ∏_∈ S_x^_γ()) and u⃗∈(∏_∈ S_y^_γ()). Denote by ^x_γ the product ∏_∈ S_x^_γ(), and similarly ^y_γ. So, 

    [γ  is sampled]=∑_v⃗∈(^x_γ)
    u⃗∈(^y_γ)|⟨v⃗|u⃗⟩ |^2/n.

By the fact that B_x is a mutual orthonormal basis of eigenvectors for ρ(S_X), every v⃗∈ B_x is either in the image of ^x_γ or in its kernel. Therefore, the above sum can be written as 

    ∑_v⃗∈(^x_γ)
    u⃗∈(^y_γ)|⟨v⃗|u⃗⟩ |^2/n=∑_v⃗∈ B_x
    u⃗∈ B_y|⟨^x_γv⃗|^y_γu⃗⟩ |^2/n=∑_v⃗∈ B_x
    u⃗∈ B_y|⟨^y_γ^x_γv⃗|u⃗⟩ |^2/n.

Since B_y is an orthonormal basis and ^x_γ,^y_γ are orthogonal projections,  for every v⃗∈ B_x we have 

    ∑_u⃗∈ B_y|⟨^y_γ^x_γv⃗|u⃗⟩ |^2=^y_γ^x_γv⃗^2=⟨^y_γ^x_γv⃗|^y_γ^x_γv⃗⟩ =⟨^x_γ^y_γ^x_γv⃗|v⃗⟩.

Thus,

    ∑_v⃗∈ B_x
    u⃗∈ B_y|⟨^y_γ^x_γv⃗|u⃗⟩ |^2/n=∑_v⃗∈ B_x⟨^x_γ^y_γ^x_γv⃗|v⃗⟩/n=τ(^x_γ^y_γ^x_γ)=τ(^x_γ^y_γ),

where the last equality uses the cyclicity of the trace and the fact ^x_γ is a projection. All in all, 

    [γ  is sampled]=τ(^x_γ^y_γ),

as needed.


 
The formula (<ref>) for the probability a specific γ is sampled is motivated by the Born measurement rule from quantum mechanics. In quantum mechanics, a (projective) measurement is specified by choosing a PVM  A→ B(ℋ) acting on a separable Hilbert space ℋ (Definition <ref> provided the finite dimensional case). 
Given a quantum state (unit vector) ψ∈ℋ, a measurement of ψ returns the outcome a with probability 
    [a   is the  measurement outcome]=ψ^* _a ψ.
 
If there are two systems (e.g. two “players”), then a Hilbert space ℋ_A, ℋ_B is associated to each of them
respectively, such that the joint Hilbert space is ℋ=ℋ_A ⊗ℋ_B. 
The joint distribution on outcomes is then given by probabilities 

    [(a,b)  are the  measurement outcomes]=ψ^* (_a^x ⊗ (^y_b)^†) ψ,

where † is transposition with no complex conjugation, as opposed to *.[It is common to drop the transposition † on the ℋ_B POVM in (<ref>). But, in this case, a transposition will appear in the calculation using traces (<ref>). So, we decided on this equivalent formulation.]
The case where ℋ_A and ℋ_B have the same (finite) dimension n, and ψ= n^-1/2∑_i e_x,i⊗ e_y,i for orthonormal bases {e_x,i} and {e_y,i} of ℋ_A and ℋ_B respectively, termed “maximally entangled,” yields 

    [(a,b)  are the  measurement outcomes]=τ(^x_a ^y_b).
 
 


    The  value of the strategy ρ against the non-local game  is

    (,ρ)   =_xy∼μ_γ∼ρ[D_xy(γ)]
       =∑_xy∈ E∑_γ S_xy→_2μ(xy)D_xy(γ)τ(∏_∈ S_x^_γ()∏_∈ S_y^_γ()).

 The (synchronous) quantum value of , ^*(), is the supremum over all possible strategies against .
 




The main result in <cit.> is a reduction from the Halting Problem to approximating  the quantum value of a game:

There exists a polynomial time algorithm that takes as input (the encoding of) a Turing machine M and outputs (the encoding of) a non-local game _M such that:

    
  * Sampling xy∈ E according to  μ and evaluating  D_xy(·) from the encoding of the  game _M  can be done in time poly(|M|), where |M| is the bit-length of the encoding of M.
    
  * If M halts, then there exists a quantum strategy ρ that commutes along edges, for which  (_M,ρ)=1. In particular, ^*(_M)=1.
    
  * If M never halts, then ^*(_M)≤1/2.


The  goal of the next section is to show that a somewhat stronger variation of Theorem <ref>, which is proved in <cit.>,  provides a negative solution to the Aldous–Lyons Conjecture <ref>.



 §.§ Permutation strategies

Our goal in  Section <ref> is to relate values of subgroup tests to values of non-local games and vice versa. The strategies for subgroup tests are finitely described IRSs, which are (induced by) maps of the form σ S→(n), while quantum strategies for non-local games are maps of the form ρ S→ U(n) adhering to certain conditions. These already seem quite related, as embedding (n) into U(n) as permutation matrices is natural, and indeed transforms every (nice enough) finitely described IRS into a quantum strategy. But, there is a problem with doing this naively: 
Say that we want to construct a quantum strategy, such that  for a certain ∈ S, regardless of anything else,  γ() will always be 1. Then, we can achieve it by choosing ρ()=-. But, - is not a permutation matrix. Therefore, any σ that will be naively transformed into a quantum strategy by embedding (n) into U(n) as permutation matrices cannot ensure a specific  is always evaluated to 1. 
To amend that, we use a tool developed in the study of solution groups of linear constraint system games (cf.  <cit.>): 
We add a special generator  that plays the role of -. 
Since - is a central involution without fixed points, the image of  under σ needs to behave the same. 
This forces σ to act on an even sized set. 
Now, if we want σ()=σ() to translate into γ() being always 1, we will need to restrict ourselves to the (-1)-eigenspace of σ(). This leads us to the following two definitions. 


    A permutation strategy for a non-local game  is a map 
σ S∪{}→(2n)
with the following properties:

    
  * The image of  is a central involution with no fixed points. Namely, 
    
    σ()^2=,  d_H(σ(),)=1,
       ∀∈ S   σ()σ()=σ()σ().

    
  * The rest of the images are involutions as well. Namely,
    
    ∀∈ S   σ()^2=.

    
  * The image of S_x commutes. Namely, for every fixed vertex x∈ V, 
    
    ∀,'∈ S_x   σ()σ(')=σ(')σ().





    Let σ S∪{}→(2n) be a permutation strategy for . Embed (2n) in U(2n) as permutation matrices. Note that in this viewpoint, U(2n) is acting naturally on functions f [2n]→ℂ, and the matrices are represented with respect to the standard basis, which consists of the indicators 
    
    ∀⋆∈ [2n]    1_⋆(♢)=
            1    ♢=⋆,
    
            0    ♢≠⋆.

    Moreover, for a permutation ζ∈(2n) and function f [2n]→ℂ, we have
    
    ∀⋆∈ [2n]   (ζ.f)(⋆)=f(ζ^-1.⋆).

    By (1) of Definition <ref>, σ() is a fixed point free involution, and thus its (+1)-eigenspace W^+ and (-1)-eigenspace W^- are of the same dimension n. Note that W^+ consists of all functions f [2n]→ℂ that are constant along -labeled edges in the Schreier graph[Generalized Schreier graphs were defined in  Section <ref>.] (σ,S∪{}), while W^- consists of all functions that flip their signs along -labeled edges. Let =^_1ℂ^2n→ W^- be the orthogonal  projection on W^-. Note that ^*=^_1 and ^*=_W^-.
    
    We can choose bases for W^+ and W^- as follows. 
    If we denote 
    
    ⋆'=σ().⋆

    for every ⋆∈ [2n], then ⋆⋆' is the perfect matching induced by -labeled edges in  (σ,S∪{}). 
    So, an orthogonal basis of W^+ may be taken to be  B^+={ 1_⋆+ 1_⋆'}_⋆∈ [2n], while for W^- one takes  B^-={ 1_⋆- 1_⋆'}_⋆∈ [2n] —  B^- contains for each vector also its opposite, so this is a basis when choosing a representative for each such pair. The union of these bases is an orthogonal basis for ℂ^2n. Furthermore, we later refer to B^- as the standard basis of W^-≅ℂ^n.



    Given that ρ S∪{}→(2n) is a permutation strategy, there are n orbits of σ(), and σ acts on them in a well-defined manner. If we arbitrarily denote each orbit as {+⋆,-⋆}, instead of {⋆,⋆'} a la (<ref>), then σ acts naturally on the set Ω_±={±}×Ω in a way that commutes with the sign flip. Hence, a permutation strategy can be defined as a map into signed permutations, where  is mapped to the sign flip, and such that the variables at each vertex are mapped to commuting involutions. This point of view may be helpful for understanding some of the next claims and remarks. We decided not to use the language of signed permutations in this paper, though in the companion paper <cit.> it is more pronounced. 



 Let σ S∪{}→(2n) be a permutation strategy for . Recall from Remark <ref> that W^- is the (-1)-eigenspace of σ(), and that ℂ^2n→ W^- is the orthogonal projection on W^-. Then,
the quantum strategy ρ S→End(W^-) induced by σ is  the mapping to the W^--corners
    
    ρ()=σ()^*.





    Let  σ be a permutation strategy. Then, the quantum strategy ρ induced by σ, as in Definition <ref>, is indeed a quantum strategy for , as in Definition <ref>.


    The criterion for being a quantum strategy as in Definition <ref> is to be a map into unitaries such that the image of the generators S_x associated with any specific vertex commute, and all the images are involutions.
First,
 
    ρ()^*=(σ()^*)^*=σ()^* ^*=σ()^-1 ^*.

    Now,  the fact that σ() commutes with  σ() for any ∈ S implies that ^*σ()=σ()^*. Therefore, 
    
    ρ()^*ρ()   =σ()^-1^*σ()^*
       =σ()^-1σ()__2n^*^*__W^-
       =^*=_W^-,

   and the image of ρ is contained in U(W^-). Item (2) of Definition <ref> states that  σ()^-1=σ() for all ∈ S,  and combined with (<ref>) gives
   
    ρ()^*=σ()^-1^*=σ()^*=ρ(),

   i.e., the images of ρ are involutions. Finally, since σ(S_x) commutes for every fixed x∈ V, we have
   
    ∀,∈ S_x   ρ()ρ()   =σ()^*σ()^*
       =^*σ()σ()^*
       =^*σ()σ()^*
       =σ()^*σ()^*
       =ρ()ρ(),

   which finishes the proof.



[Classical permutation strategies]
Let (2)={,ζ} be the group of permutations acting on two elements. For every fixed f S→{0,1}, we can define an action σ S∪{}→(2) as follows

    σ()=ζ  ;  σ()=    f()=0,
    ζ    f()=1.

It is straightforward to see that the quantum strategy associated with σ satisfies ρ()=(-1)^f(), and thus γ S_xy→{0,1} that is sampled according to ρ is deterministically f|_S_xy.
Strategies  of this form are usually called deterministic. By taking direct sums of such actions (for potentially different f's), we can get any (rational) distribution over deterministic strategies. These strategies are usually called classical. So, every (rational) classical strategy can be obtained as a permutation strategy. 





    Let σ S∪{}→(2n) be a permutation strategy, and ρ the associated quantum strategy. Then, if for a word w∈(S∪{}) we have ρ(w)=, then σ(w)=.


    Recall from Remark <ref> that the images of ρ act naturally, and in the same way as σ,  on W^-, the (-1)-eigenspace of σ(), which consist of functions f [2n]→ℂ that flip along -edges. Recall also the notation ⋆'=σ().⋆ from (<ref>).
    So, the fact that ρ(w)= means that for any  function f∈ W^-, σ(w).f=f as well.  
    This is true in particular for B^-={ 1_⋆- 1_⋆'}_⋆∈ [2n], the basis of W^- described in the same remark. Note that for every ζ∈(2n) we have ζ. 1_⋆= 1_ζ.⋆.
    Hence, for every ⋆∈[2n], we have

    1_⋆- 1_⋆'=σ(w).( 1_⋆- 1_⋆')= 1_σ(w).⋆- 1_σ(w).⋆'.

By evaluating the functions on both sides of the above equation on the point ⋆∈ [2n], we get

    1= 1_σ(w).⋆(⋆)- 1_σ(w).⋆'(⋆),

which means σ(w).⋆=⋆. Since ⋆ was any point in [2n], σ(w)= and we are done.





     A non-local game  is said to have a perfect permutation strategy if there exists a permutation strategy σ such that the  quantum strategy ρ induced by it, as in Definition <ref>, satisfies (,ρ)=1.
 


In the rest of this subsection we further analyze permutation strategies, and specifically discuss a procedural sampling — similar to  Claim <ref>  — of a γ S_xy→{0,1} according to ρ which is induced by a permutation strategy σ. 



 §.§ Interlude — Fourier bases of _2^m actions



    Let σ_2^k→(_2^k) be the standard action of _2^k on itself, namely σ(w).v=v+w for every v,w∈_2^k. The Fourier basis of σ is the following collection of functions, denoted by _2^k: For every u∈_2^k, let u_2^k→ℂ be 

    ∀ w∈_2^k   u(w)=(-1)^⟨ u,w⟩/√(2^k),

where ⟨ u,w⟩=∑_i=1^k u_iw_i is the standard bilinear form on _2^k.[We try to keep ⟨·,·⟩ for the bilinear form over a finite vector space, and ⟨· |·⟩ for the complex inner product.]



    The collection _2^k is an orthonormal basis of eigenvectors for the action  σ. 


    The action σ acts as permutations on the standard basis of ℂ^_2^k, and thus extends to act on the whole vector space of functions from _2^k to ℂ. Using the standard inner product ⟨·|·⟩ on  ℂ^_2^k as before, we can check that 
    
    ∀ u∈_2^k   ⟨u|u⟩ =∑_w∈_2^ku(w)·u(w)=∑_w∈_2^k1/2^k=1.

    Furthermore, 
    
    ∀ u≠ v∈_2^k   ⟨v|u⟩   =∑_w∈_2^kv(w)·u(w) 
       =1/2^k∑_w∈_2^k(-1)^⟨ v+u,w⟩
       =0,

    where the last equality is derived from the fact v+u≠0⃗. Lastly, 
    
    ∀ u,v,w∈_2^k   (σ(w).v)(u)   =v(u+w)
       =(-1)^⟨ u+w,v⟩/√(2^k)
       =(-1)^⟨ w,v⟩·(-1)^⟨ u,v⟩/√(2^k)
       =(-1)^⟨ w,v⟩v(u),

    and _2^k is indeed a collection of mutual eigenvectors for σ.


Recall basic group actions definitions. 
Let G be a group, and σ G→(X),τ G→(Y) be two homomorphisms, which induce actions of G on X and Y respectively. A map 
f X→ Y  is a factor (of G-actions)  — also known as a morphism of G actions or a  (σ,τ)-equivariance — if
for all w∈ G and x∈ X, we have f(σ(w).x)=τ(w).f(x).
Note that for every set S⊆ G, such a map induces a morphism of oriented edge-labeled graphs between (σ,S) and (τ,S).
The actions σ and τ are said to be equivalent if there exists a bijective factor between them, which in the graph case would be a (edge-labeled oriented) graph isomorphism between (σ,S) and (τ,S).
Given two actions, we can define their sum  σ⊕τ G→(X⊔ Y), where  ⊔  is the  disjoint union, to be

    ∀ w∈ G, z∈ X⊔ Y   
     (σ⊕τ(w)).z=σ(w). z     z∈ X,
    τ(w). z     z∈ Y.


and their  product σ⊗τ S→(X× Y) to be

    ∀ s∈ S, x∈ X,y∈  Y   (σ⊗τ(s)).(x,y)=(σ(s).x ,τ(s).y).


If σ is a transitive action and  x∈ X, then it is equivalent to the  action of G on G/(σ,x), where (σ,x) is defined (as in Section <ref>) to be
(σ,x)={w∈ G|σ(w).x=x}.
The bijective factor in this case is associating with σ(w).x the coset w·(σ,x), and recalling that by transitivity every y∈ X is of the form σ(w).x for some w∈ G. So,  every action σ G→(X) is equivalent to the sum of actions of G on quotients of it G/H, for some collection of subgroups H≤ G. 

Assume now that G is _2^m for some positive integer m. Let σ G→(X) be a transitive action. Since every subgroup H of  G is normal, this action factors through  some _2^k for k≤ m. Namely, we can find a bijective factor between X and the action of _2^k on itself: 
 Fix some ⋆∈ X; (σ,⋆) is a subspace of _2^m, and we implicitly assumed it is isomorphic to _2^m-k; for every ♢∈ X, there is an element g of _2^m for which σ(g).⋆=♢, and thus σ(g').⋆=♢ for every g'∈ g+(σ,⋆); so we can associate g+(σ,⋆) with ♢ and _2^m/(σ,⋆) with _2^k.  
For every g∈_2^m, let g̃∈_2^k be its image in the bijection between the quotient space _2^m/(σ,⋆) and _2^k. Then,  

    ∀ v∈_2^k   σ(g).v=(-1)^⟨ v,g̃⟩v.

In particular, for s∈ S⊆_2^m, the action of s on v can be read from what happens along edges labeled by s in (σ,S) — if v(⋆)=v(σ(s).⋆), then v is in the (+1)-eigenspace of σ (s), and otherwise it is in its (-1)-eigenspace.

All in all, given a not necessarily transitive action σ of _2^m on X, its orbits have Fourier bases as in Definition <ref> (extended to being zero on all vertices outside of the orbit), and  the disjoint union of these will be called  the Fourier basis of σ. 
Note that since the bijections between the orbits and _2^k can be chosen in many ways, the Fourier basis of σ is determined only up to the sign of each basis vector. 





 §.§ Procedural sampling according to permutation strategies

Let σ S∪{}→(2n) be a permutation strategy for , and recall Remark <ref> and the notation therein. 
By Definition <ref>, σ(S_x∪{}) consists of commuting involutions. 
Therefore, the group generated by σ(S_x∪{}) is a quotient of _2^S_x∪{} — this can be acquired by linearly extending the map 1_↦σ() for every ∈ S_x∪{}, where 1_S_x∪{}→_2 is the indicator of . 
This means that 

    σ_x=σ|_S_x∪{}
 
induces an action of  _2^S_x∪{} on [2n]. 
Choose B_x ⊂ℂ^2n to be the Fourier basis of σ_x, as was defined in the Interlude Section <ref>.
Let B_x^-= B_x∩ W^-⊂ B_x be the collection of n eigenvectors in B_x on which σ() acts as (-1), namely all v⃗∈ B_x such that  σ().v⃗=-v⃗. These are the Fourier basis elements that flip along edges labeled by  in (σ_x,S_x∪{}). Then, B_x^- is an orthonormal basis of eigenvectors for ρ(S_x).


    Let  σ S∪{}→(2n) be a permutation strategy for , and ρ S→ U(W^-) the quantum strategy associated to σ. Then, the sampling procedure of γ S_xy→{0,1} according to ρ can be described  as follows:
    
        
  * Sample ⋆∈ [2n] uniformly at random. Then, it is part of an orbit O_x⊆ [2n] of σ_x (defined in (<ref>)) and O_y⊆ [2n] of σ_y.
        
  * Choose a pair v⃗∈ B^-_x and u⃗∈ B^-_y such that v⃗ is supported on O_x and u⃗ is supported on O_y, with probability 
        
    2⟨v⃗|u⃗⟩ ^2/|O_x∩ O_y|.

        
  * For ∈ S_x, let γ()=0 if ρ().v⃗=v⃗ and 1 otherwise. Similarly, for ∈ S_y, let γ()=0 if ρ().u⃗=u⃗ and 1 otherwise.
    
    We may use the notation γ∼σ instead of γ∼ρ to refer to sampling according to a permutation strategy.



The probability of sampling ⋆∈ [2n] which is in O_x∩ O_y is |O_x∩ O_y|/2n. Thus, the probability of sampling a specific pair v⃗∈ B_x^-,u⃗∈ B_y^- whose intersection of supports is O_x∩ O_y is
    
    |O_x∩ O_y|/2n·2⟨v⃗|u⃗⟩^2/|O_x∩ O_y|=⟨v⃗|u⃗⟩ ^2/n.

    This shows that the above sampling procedure provides the same distribution as the one in Claim <ref>.

    IS THIS PROOF REDUNDANT.
    We mainly need to show that the above is genuinely a sampling scheme, which boils down to the second step defining a probability distribution, i.e.,
    
    ∑_v⃗∈ B_x^-Supp(v⃗)=O_x 
    u⃗∈ B_y^-Supp(u⃗)=O_y⟨v⃗|u⃗⟩ ^2=|O_x∩ O_y|/2.

    This is because, once this is true, 
    the probability of getting a specific pair is still ⟨v⃗|u⃗⟩ ^2/n. To see that, 
    note that  the probability of sampling ⋆∈ [2n] which is in O_x∩ O_y is |O_x∩ O_y|/2n, and thus the probability of sampling a specific pair v⃗,u⃗ whose intersection of  supports is O_x∩ O_y is
    
    |O_x∩ O_y|/2n·2⟨v⃗|u⃗⟩^2/|O_x∩ O_y|=⟨v⃗|u⃗⟩ ^2/n.

    Note that if v⃗ and u⃗ do not have a common support, then their inner product is zero and the calculation remains valid.

Let 1_O_x be the indicator of O_x (it is actually a scalar multiple of a basis element in B_x, though not in B_x^-). Let u⃗· 1_O_x be the restriction of u⃗ to O_x, namely the function which agrees with u⃗ on O_x, and outputs 0 everywhere else. 
Since u⃗ is a Fourier basis vector supported on O_y, its non-zero entries are all of absolute value 1/√(|O_y|). Thus, ⟨u⃗· 1_O_x|u⃗· 1_O_x⟩=|O_x∩ O_y|/|O_y|. 
On the other hand, u⃗· 1_O_x is supported on O_x. 
Moreover, since  preserves O_x and O_y (by their definition), whenever ⋆∈ O_x∩ O_y, also σ().⋆∈ O_x∩ O_y. 
Thus, u⃗· 1_O_x is a function that flips along  labeled edges in (σ,S∪{}). All in all, we have 

    u⃗· 1_O_x=∑ _v⃗∈ B_x^-Supp(v⃗)=O_x ⟨v⃗|u⃗· 1_O_x⟩_⟨v⃗|u⃗⟩v⃗,

    and in particular 
    
    u⃗· 1_O_x^2=∑ _v⃗∈ B_x^-Supp(v⃗)=O_x ⟨v⃗|u⃗⟩^2.

    Summing up over the |O_y|/2 many u⃗∈ B^-_y which are supported on O_y, we deduce the claim.











 §.§ Tailored games

 As permutation strategies allow us to translate finitely described strategies of subgroup tests to quantum strategies of non-local games, the soon to be defined Z-aligned permutation strategies of tailored non-local games will allow us to move in the other direction. It is harder to motivate our choices beforehand in this case, but we hope that Section <ref>  clarifies what may seem cryptic in this section. We begin by restricting the family of non-local games that we focus on — for the reader familiar with linear constraint system games, we note that this family is a generalization of them.
 

Colloquially,  a tailored non-local game is one where D_xy reads part of γ, and decides according to this partial view which parity checks to apply on the whole of γ.[We considered calling such games controlled linear, since it is more informative. But, since conditionally linear is a term used in MIPRE,Tailored_MIPRE, and terms containing linear are generally overused, we decided to use a less informative notion.]

Formally,
   a tailored non-local game  is equipped with an extra structure, described shortly, and its decision functions D_xy behave canonically with respect to this extra data.
    Instead of a single length function ℓ,  has two length functions ℓ^ V→ℕ and ℓ^ V→ℕ, and ℓ=ℓ^+ℓ^. Before, the length function described the size of the formal generating set at each vertex. 
    Now, the formal set of generators S_x at x∈ V will be a disjoint union of the sets S_x^ and S_x^, where S_x^={^x,,i| 1≤ i≤ℓ^(x)} and S_x^={^x,,i| 1≤ i≤ℓ^(x)}. The elements of  S_x^ are  called readable variables at x∈ V and the elements of S_x^ the linear or unreadable variables at x. 
    Let S→{0,1} be the readability function, namely, the indicator of whether ∈ S is  readable or not.
      In addition,  is equipped with  a collection of controlled linear constraints functions L_xy that take as input a function γ^ S_x^∪ S_y^→_2, and outputs a collection of subsets of S_xy∪{}. Namely,
    
    L_xy_2^S_x^∪ S_y^→_2^_2^S_xy∪{}.

    The image of L_xy is interpreted as a collection of linear constraints that will be verified by the decision function.   
    Finally, the decision function D_xy(γ) behaves as follows: It restricts γ to the readable variables, namely looks at γ^=γ|_S_x^∪ S_y^ S_x^∪ S_y^→_2, and calculates L_xy(γ)=L_xy(γ^). Then, it extends γ such that γ()=1. Finally, for every  α∈ L_xy(γ), we have α S_xy∪{}→_2,  and D_xy verifies that 
    
    ⟨α,γ⟩=∑_∈ S_xy∪{}α() ·γ()=0.
 
    Namely, L_xy(γ) consists of linear constraints that γ needs to satisfy. 



    Though being tailored  seems to be quite a restrictive form for a non-local game, every game can be tailored in a trivial manner. First, all variables are declared readable, namely ℓ^=ℓ and ℓ^=0. Then, if the decision function D_xy decided to accept γ according to the original game, then it lets L_xy(γ) be empty (and thus all linear conditions will be satisfied regardless of what γ is). And, if D_xy decided to reject γ according to the original game, 
    then it chooses L_xy(γ) to contain the singleton {} as the single subset appearing in  L_xy. Note that {} represents the linear equation 1·γ()=0, which is 1=0, and thus cannot be satisfied. 


The last remark raises the question: What have we gained by defining tailored non-local games, if any game can be tailored in a straightforward manner? 


   A permutation strategy σ for a tailored non-local game  is said to be Z-aligned if the assignments  σ() for readable variables act on each point in [2n] either like the identity  or like σ(). This is equivalent to ρ() being a diagonal matrix when presented according to  the standard basis B^- of W^-.[ The standard basis is usually called the Z-basis in quantum information theory, and thus the name Z-aligned strategies.]  



    The classical permutation strategies described in Example <ref> are Z-aligned. But, one can construct permutation strategies that induce a classical strategy in the usual sense (i.e., whose images are all commuting) without being Z-aligned. 

It is clearer now why the way one tailors a non-local game matters: The existence of a perfect Z-aligned permutation strategy for the game  depends on it. Let us demonstrate this with binary linear constraint system (LCS) games, and specifically with the magic square game. For  a full description of  the magic square game we refer to <cit.> or <cit.>, and for a general  introduction to LCS games see <cit.>. 

[Linear constraint system games]
   Let A be an m× n matrix with _2 coefficients, and let b⃗ be a column vector in _2^m. Classically, such a pair defines a system of linear equations Ax⃗=b⃗ over _2.  It also defines a certain non-local game (A,b⃗) which is the quantum counterpart of this classical system of equations. Let us provide a dramatization of this game, in the spirit of Remark <ref>. In  (A,b⃗), the referee chooses a random linear constraint in Ax⃗=b⃗ (i.e., a row) and sends it to one of the players, and chooses  a random variable that appears  in the chosen constraint (i.e., a column whose intersection with the chosen row is non-zero) to the other player. It then expects the row player to respond with an assignment to the variables appearing in this constraint, and from the column player to respond with an assignment to its single variable. The referee declares this to be a winning condition if the row player's assignment satisfied the constraint, and both players agree on the value of the variable given to the column player. 
   The column player is thought to hold a global assignment and answer according to it, while the row player is assumed to have a satisfying assignment to each constraint — intuitively, by  checking consistency between the players, the referee verifies that the global assignment of the column player indeed satisfies all the constraints. 
   
   Let us define this game rigorously. 
   The vertices in the underlying graph of (A,b⃗) will be indexed by the rows and columns of the matrix A, namely {r_i| i∈ [m]} and {c_j| j∈ [n]}. There is an edge between a row r_i and a column c_j if and only if A_ij=1. The length of every column vertex is 1, and we denote by _j the variable associated with the j^ th column c_j. The length of each row vertex is the number of 1's in the row, and we associate variables S_r_i={_ij'| A_ij'=1} to every row r_i. The decision function D_r_ic_j gets as input an assignment γ to _j and {_ij'| A_ij'=1}, and accepts if and only if 

    ∑_j' A_ij'=1γ(_ij')=b_i   and  γ(_j)=γ(_ij).

Though for our discussion the distribution μ over edges in this game is not important, one can consider the following standard sampling scheme: 1) Choose a row uniformly at random. 2) Choose a uniform column out of the support of the chosen row.

Let us describe a non-trivial tailoring of  (A,b⃗). First, all variables are unreadable, namely ℓ^=0 and ℓ^=ℓ. Given that the edge r_ic_j was sampled, the system L_r_ic_j will consist of two checks, which are derived from 
(<ref>):[note that since there are no readable variables, L_r_ic_J is constant.] 

    α_ consistency()   =
        0    ≠_j,_ij
    
        1    otherwise
    α_ linear()   =
        0    =_j
    
        1    =_ij'∈ S_r_i
    
        b_i    =

Then, α_ consistency will force D_r_ic_j to check that γ(_j)=γ(_ij), and α_ linear will force it to check that ∑_j' A_ij'=1γ(_ij')=b_i, and thus it acts the same way as before. 


   The difference between the above tailored form of (A,b⃗) and the one suggested in Remark <ref> may seem technical. But, here all the variables are unreadable, and in the version of Remark <ref> version all variables are readable. 
   If all variables of a tailored game are readable, for it to have a perfect  Z-aligned permutation strategy is the same as having a perfect classical strategy — which in turn is the same as for the linear system Ax⃗=b⃗ to have a solution. 
   But, when all the variables are unreadable, there could be a perfect Z-aligned permutation strategy without Ax⃗=b⃗ having a solution. 
   For example, the perfect strategy for the Mermin–Peres magic square game (see for example <cit.>) can be derived from a permutation strategy (see our companion paper <cit.>), and since there are no readable variables in this case,  it is automatically Z-aligned.
   Furthermore, the magic square game has no perfect classical strategy. This shows that if we care about perfect Z-aligned permutation strategies, the way we tailor a game matters.







§ MAIN THEOREM II: TESTS ASSOCIATED WITH TAILORED GAMES


The goal of this section is to define a mapping from tailored non-local games to subgroup tests such that: 

    
  * Completeness: Perfect Z-aligned permutation strategies that commute along edges for the tailored game are translated to perfect finitely described strategies of the subgroup test.
    
  * Soundness: Almost perfect finitely described strategies for the subgroup test are close to almost perfect Z-aligned permutation strategies for the tailored non-local game. 
 

Throughout this section,  is a tailored non-local game with underlying graph G=(V,E),  length functions ℓ^,ℓ^ V→ with ℓ=ℓ^+ℓ^ and Λ =max_x∈ V(ℓ(x)), formal sets of generators S_x=S_x^∪ S_x^ for every vertex x∈ V, where S_x^={^x,,i| 1≤ i≤ℓ^(x)} and S_x^={^x,,i| 1≤ i≤ℓ^(x)}, a readability function S→{0,1} where S=⋃_x∈ V S_x, a distribution μ over  edges E, and decision functions D_xy_2^S_xy→_2 for every edge xy∈ E with a controlled linear constraints map L_xy_2^S_x^∪ S_y^→_2^_2^S_xy∪{}. 




    The synchronous subgroup test =() associated with the tailored non-local game  is defined as follows: The set of generators S̃ in the test  is S∪{}. The set  Q̃ which parametrizes the challenges of  is the edge set E of , and the distribution μ̃ over challenges is μ of . The decision function D̃_xy works as follows (and the subsets K̃_xy can be derived from this description): Given a subgroup H of (S̃), 
    
        
  *  It verifies that
    
    ∉ H  ;  ^2∈ H  ;  ∀∈ S_xy   [,]∈ H.

    
  *  It verifies that
    
    ∀∈ S_xy      ^2∈ H,
    ∀,'∈ S_x      [,']∈ H,
    ∀,'∈ S_y      [,']∈ H.

        
  *  
    It verifies that 
    
    ∀∈ S_x^∪ S_y^   ∈ H  or  ∈ H.

        
  *  If H passed all the previous checks, then the following function γ̃^ S_x^∪ S_y^→_2 can be calculated: γ̃^()=0 if ∈ H and γ̃^()=1 if ∈ H. After recovering this γ̃^, the subset 
        
    L̃_xy(H)=L_xy(γ̃^)

        can  be calculated.
        So, the decision function verifies that 
        
    ∀α∈L̃_xy(H)   ^α()·∏_∈ S_x^α()·∏_∈ S_y^α()∈ H,

          where the product is according to some pre-fixed ordering of S_x and S_y.
    
  
    If any of the above checks did not go through, D̃_xy rejects H, and otherwise it accepts it.



    Let us motivate the checks that the decision function D̃_xy is applying on H. Let σS̃→(X) be a finitely described strategy. If σ always passes clause:Check_1_in_associated_test, it means that  acts as a fixed point free central involution on X, which is part of the requirement of a permutation strategy  for  (see Definition <ref>). If σ always passes clause:Check_2_in_associated_test, then the images of all other variables in S̃ are involutions, and for any fixed x∈ V, σ(S_x) commutes, which are again requirements for σ to be a permutation strategy of . If σ always passes clause:Check_3_in_associated_test, then the readable variables always act as either  or , which implies σ is a Z-aligned permutation strategy for  (see Definition <ref>). So, the goal of the first three Checks is to force any finitely described strategy of  to be a permutation Z-aligned strategy for . Lastly, clause:Check_4_in_associated_test verifies “the same”  linear relations as , which means that the winning probability of σ  against   and  are similar.


We are ready to formulate our main theorem:

    Let  be a tailored non-local game with max_x∈ V (ℓ(x))=Λ>0, and let =() be its associated synchronous subgroup test, as in Definition <ref>. 
    
        
  *  Completeness: If  has a perfect Z-aligned permutation strategy that commutes along edges, then  has a perfect finitely described strategy.
        
  *  Soundness: If  has a finitely described strategy σ with (,σ)≥ 1-, then there exists a quantum strategy ρ such that (,ρ)≥ 1-CΛ ^4· 2^6Λ, where C>0 is a universal constant.
    

The next section is devoted to the proof of Theorem <ref>.  


    There exists a polynomial time algorithm that takes as input (the encoding of) a Turing machine M and outputs (the encoding of) a tailored non-local game _M such that:

    
  *  Sampling xy∈ E according to  μ and evaluating  D_xy(·) from the encoding of the  game _M  can be done in time poly(|M|), where |M| is the bit-length of the encoding of M.
    
  *  If M halts, then _M has a perfect  Z-aligned permutation strategy that commutes along edges.
    
  *  If M never halts, then ^*(_M)≤1/2.


Theorem <ref> is proved in <cit.>.

   The Aldous–Lyons Conjecture <ref> has a negative solution.


The idea is as follows. Given a Turing machine M, we can define  the tailored non-local game  _M calculated in Theorem <ref>, and then calculate the associated subgroup test (_M)  from Definition  <ref>. In the next paragraph, the following two facts are proved:

    
  * If M halts, then (_M) has sofic value 1.
    
  * On the other hand, if M does not halt, then the sofic value of (_M) is smaller than 1-λ(M), where λ(M)=2^-c|M|^c for some universal constant c>0 independent of M. 
 
 Recall that by Theorem <ref>, there is a computable sequence approaching _((_M)) from above and another computable sequence approaching _((_M)) from below. If the Aldous–Lyons Conjecture <ref> has a positive solution, then these values are the same, and as Corollary <ref> states, one can approximate _((_M)) up to any predetermined additive error θ. As λ(M) can be computed directly from |M|, one can choose θ=λ(M)/2, and thus deduce whether _((_M))<1 or _((_M))>1-λ(M). This in turn allows one to decide whether M halts or not. Since the Halting Problem is undecidable, this is a contradiction, which implies that Conjecture  <ref> must have a negative solution.



Let us prove the two bullets above. Since the running time of the verification procedure of _M bounds Λ=max_x∈ V(ℓ(x)), we can use clause:1_tailored_MIP* of Theorem <ref> to deduce that Λ=poly(|M|). 
    If M halts, then by clause:2_tailored_MIP* of Theorem <ref> and clause:1_in_values_associated_test of Theorem <ref>, we have _((_M))=1.
    If M does not halt, then by clause:3_tailored_MIP* of Theorem <ref>, we have ^*(_M)≤1/2. Hence, by clause:2_in_values_associated_test of Theorem <ref>,
    we have
    
    1/2≥ 1-CΛ^4· 2^6Λ(1-_((_M))).

    Rearranging this inequality, we get 
    
    _((_M))≤ 1-1/2CΛ^4· 2^6Λ.

    The quantity 1/2CΛ^4· 2^6Λ can be bounded from below by λ(M)=2^-c|M|^c, where c>0 is a universal constant that depends on the constants implicit in the notation  Λ=poly(|M|) (guaranteed by Theorem <ref>), as well as the constant C>0 appearing in Theorem <ref>. This finishes the proof.







§ PROVING THEOREM <REF>






 §.§ Proving the completeness in Theorem <ref>


This is the easy direction, we have done most of the work towards it, and here we use the notation that we have established before. Recall that S̃=S∪{}. Let σ S∪{}→(2n) be a perfect Z-aligned permutation strategy that commutes along edges for the tailored non-local game . Further recall that the finitely described IRS Φ(σ) samples a subgroup H of  as follows — it takes a uniform ⋆∈ [2n] and outputs H=(σ,⋆). By Definition <ref>,

    d_H(σ(),)=1   ;   σ()^2=  ;  ∀∈ S   σ()σ()=σ()σ().

Therefore, for every ⋆∈ [2n], 

    ∉(σ,⋆)   ;  ^2∈(σ,⋆)  ;  ∀∈ S   [,]∈(σ,⋆),

and when Φ(σ) runs against  it  always passes clause:Check_1_in_associated_test. Also by  Definition <ref>,

    ∀∈ S σ()^2=  ;  ∀ x∈ V, ∀,'∈ S_x   σ()σ(')=σ(')σ().

Therefore, for every ⋆∈ [2n],

    ∀∈ S ^2∈(σ,⋆)  ;  ∀ x∈ V, ∀,'∈ S_x   [,']∈(σ,⋆),

and when Φ(σ) runs against  it always passes clause:Check_2_in_associated_test.
By Definition <ref>, for every ⋆∈ [2n] we have 

    ∀∈^-1(1)   σ().⋆=⋆  or  σ().⋆=σ().⋆.

Namely, for every ⋆∈ [2n],

    ∀∈^-1(1)   ∈(σ,⋆)  or  ∈(σ,⋆),

and when Φ(σ) runs against  it always passes clause:Check_3_in_associated_test.

We are left to show that Φ(σ) always passes clause:Check_4_in_associated_test.
To that end, we need a couple of claims.
    The first is a well known and commonly used observation in quantum information theory. For the convenience of the reader, we provide a proof.

    Let S be a finite set, ρ S→ U(n) such that ρ(S) commutes[This claim can be generalized so that a weaker assumption is used. See Section 3 of <cit.>.] and all the images are involutions.  Then, given a fixed α S→_2, we have ∑_∈ Sα()γ()=0  with probability 1 when  γ∼ρ if and only if ∏_∈ Sρ()^α()=. Namely,

    _γ∼ρ[⟨α,γ⟩=0]=1 ∏_∈ Sρ()^α()=.



Assume without loss of generality that α is fully supported. Otherwise, we can focus on ρ|_Supp(α) and proceed in the same manner. 
    As before, we can write ρ()=_0^-_1^, where ^_0 is the projection on the (+1)-eigenspace of ρ(), and _1^ is the projection on its (-1)-eigenspace.  By Definition <ref>, 

    _γ∼ρ[∑_∈ Sγ()=0]   =∑_γ S→_2
    ∑γ()=0τ(∏_∈ S_γ()^).

Note that ^_0+^_1=, and thus 
∑_γ S→_2τ(∏_∈ S_γ()^) is always 1. 
Therefore, 

    τ(∏_∈ Sρ())   =τ(∏_∈ S(^_0-^_1))
       =∑_γ S→_2
    ∑γ()=0τ(∏_γ()^) -∑_γ S→_2
    ∑γ()=1τ(∏_γ()^)
       =2·∑_γ S→_2
    ∑γ()=0τ(∏_γ()^)-1
       =2·_γ∼ρ[∑_∈ Sγ()=0]-1.

Thus,  ∏ρ()= if and only if τ(∏ρ())=1 if and only if  _γ∼ρ[∑_∈ Sγ()=0]=1, which finishes the proof.



    Let σ S∪{}→(2n) be a Z-aligned permutation strategy for a tailored game , and let ρ be the quantum strategy associated with σ (as in Definition <ref>). 
    Then, the restriction of  γ∼ρ to the readable variables in S_xy (which we denoted by γ^) depends only on the vertex ⋆∈ [2n] sampled in the beginning of the procedure outlined in Claim <ref>.
    Furthermore, if we denote the sampled L_xy(γ) by L_xy(⋆), emphasizing its dependence on the sampled vertex, then L̃_xy((σ,⋆))=L_xy(⋆), where L̃_xy was defined in (<ref>).


    Let ⋆∈ [2n] be some fixed vertex, O_x be the σ_x orbit of ⋆, and ∈ S_x be a readable variable. Since σ is Z-aligned, either σ().⋆=⋆ or σ().⋆=σ().⋆. Now, as discussed in Section <ref>, σ acting on  O_x is equivalent to an action of _2^k on itself. Furthermore, checking whether  v⃗∈ B^-_x that is supported on O_x is in the (+1) or (-1) eigenspace of σ() depends only on whether v⃗ flips along an -labeled edge in (σ,S) for some vertex in O_x. Since v⃗∈ B^-_x, it must flip along -labeled edges. So, if σ().⋆=⋆ then σ()v⃗=v⃗ and γ()=0, and if σ().⋆=σ().⋆ then σ()v⃗=-v⃗ and γ()=1. 
    Since this was independent of the specific v⃗ that we considered, the first part of the claim is deduced.  
   This proves that γ̃^()=γ^(), where γ̃^() was defined in clause:Check_4_in_associated_test of Definition <ref>, and therefore L̃_xy((σ,⋆))=L_xy(γ̃^)=L_xy(γ^)=L_xy(⋆) which is the second part of the claim.


We can now come back to our proof. Let us, in the spirit of previous notations, denote  σ_xy=σ|_S_xy∪{}. Let ⋆∈ [2n], and let O_xy be the orbit of ⋆ under σ_xy. Since σ commutes along edges (see Definition <ref>), σ_xy induces an action of _2^S_xy∪{}. Since (σ_xy) is commuting, the action of readable variables in O_xy is constant, namely if a readable variable acts as  on some ⋆∈ O_xy, it will act that way on all ⋆∈ O_xy, and similarly for acting as . Furthermore, by Claim <ref>, the linear constraints sampled in the associated game L̃_xy at a specific vertex ⋆∈ O_xy are the same as the ones sampled by the tailored non-local game.
Joining these observations, there is a set of constraints L such that for every ⋆∈ O_xy we have  L̃_xy((σ,⋆))=L_xy(⋆)=L.  So, we can focus on the orbit O_xy, and study the corner of ρ, which was the quantum strategy induced by σ, with respect to the projection ^O_xyℂ^2n→ℂ^O_xy. 
Since ρ is a perfect strategy, for every γ that was sampled (according to Claim <ref>) by first choosing ⋆∈ O_xy, and for every α∈ L, we have ∑α()γ()=0. 
By Claim <ref>,  this means that ∏ρ()^α() is the identity when restricted to functions supported on O_xy that flip along -labeled edges. By Claim <ref>, we can deduce that ∏σ()^α() acts as the identity on O_xy. Since this was true for every α in L, and L=L̃_xy((σ,⋆)) for every ⋆∈ O_xy, we can conclude that σ always passes clause:Check_4_in_associated_test when run against . All in all, σ is a perfect strategy for , proving the completeness clause of Theorem <ref>.



 §.§ Proving the soundness in Theorem <ref>


The plan is as follows:

    
  * Recall that S̃= S∪{}.  Given a finitely described strategy σS̃→(n) for , with (,σ)≥ 1-, we  perturb it bit by bit until it passes with probability 1 clause:Check_1_in_associated_test, clause:Check_2_in_associated_test and clause:Check_3_in_associated_test from the definition of  (while potentially worsening the probability of passing  clause:Check_4_in_associated_test). 
    
  * Then, we can bound from below the value of this perturbed strategy using the mechanisms of Section <ref>. 
    
  * Since the resulting perturbed strategy satisfies the first three checks of , it is a Z-aligned permutation strategy for . For such strategies, we prove that their value against  induces a lower-bound on their value against . This will conclude the proof.
   

The following three propositions are the manifestation of the above plan. We abuse notations and denote by μ the marginal distribution on vertices of 𝒢, namely

    ∀ x∈ V   μ(x)=1/2∑_xy∈ Eμ(xy).

Also, recall the notion of the significance function _  associated with a test , defined in (<ref>), and the notion of the edit distance d^_ edit (<ref>) between finitely described IRSs. Finally, recall that  Λ=max_x∈ V(ℓ(x)), and assume Λ>0.

   Let σS̃→(n) be a finitely described strategy  for , with (,σ)≥ 1-. Denote by _x the losing probability in  given that an edge containing x∈ V was sampled as the challenge — note that _x∼μ[_x]=. 
   Then, there is a strategy σ'S̃→(2m), where m= ⌈n/2⌉, that passes with probability 1 clause:Check_1_in_associated_test, clause:Check_2_in_associated_test and clause:Check_3_in_associated_test in the definition of the associated test , and which is close to σ in the following sense:
   
    d_H(σ(),σ'())≤  C_0
    ∀ x∈ V, ∈ S_x      d_H(σ(),σ'())≤  C_0· 2^2Λ·Λ^3(+_x).

    for some universal constant C_0>0.



     Every  finitely described strategy 
 σS̃→(2n)   for  that passes with probability 1 clause:Check_1_in_associated_test, clause:Check_2_in_associated_test and clause:Check_3_in_associated_test in the definition of the associated test  is a Z-aligned permutation strategy for  and vice versa. Moreover, there is a universal constant C_1>0 such that  if  (,σ)≥ 1-, then (,ρ)≥ 1-C_1· 2^2Λ where ρ is the quantum strategy associated with σ (Definition <ref>). 


    Let  be the test associated with the tailored non-local game .  Then, there is a universal constant C_2>0, such that the significance _τ̃ associated with the test   satisfies
    
    _()   ≤ C_2· 2^2Λ,
    ∀ x∈ V, ∈ S_x   _()   ≤μ(x)· C_2· 2^2Λ.



    We did not try to optimise the parameters in the three preceding propositions. Actually, we decided to choose a simpler version of the associated synchronous test (Definition <ref>) that provides an exp(Λ) deterioration in the value, as seen in Proposition <ref>. We have variations of tailored games and the associated synchronous test which allow for only a poly(Λ) deterioration in the value, but we could not find any reduction that is independent of Λ.


Now, we can deduce the soundness clause of Theorem <ref> as a corollary of the preceding propositions.


    Let σS̃→(n) be the given finitely described strategy with value ≥ 1-. By Proposition <ref> and Proposition  <ref>, there is a strategy σ'S̃→(2⌈n/2⌉) passing clause:Check_1_in_associated_test, clause:Check_2_in_associated_test and clause:Check_3_in_associated_test with probability 1, and such that the significance weighted distance between σ and σ' is
    
    d^_(σ,σ')   =∑_s∈S̃_(s)· d_H(σ(s),σ'(s))
       =_()d_H(σ(),σ'())+∑_x∈ V∑_∈ S_x_()d_H(σ(),σ'())
       ≤ C_2· 2^2Λd_H(σ(),σ'())+∑_x∈ V∑_∈ S_xμ(x)· C_2· 2^2Λ d_H(σ(),σ'())
       ≤ C_0C_2·2^2Λ+∑_x∈ V|S_x|·μ(x)· C_2· 2^2Λ· C_0· 2^2Λ·Λ^3(+_x)
       ≤ C_0C_2·2^2Λ+ C_0C_2Λ ^4 · 2^4Λ(∑_x∈ Vμ(x)(+_x))__x∼μ[+_x]=2
       ≤ 3C_0C_2Λ ^4· 2^4Λ.

    Therefore, by Claim <ref>, 
    (,σ')≥ 1--3C_0C_2Λ ^4· 2^4Λ≥ 1-4C_0C_2Λ ^4· 2^4Λ.
 By applying Proposition <ref> on σ',  its associated quantum strategy ρ' satisfies  
    
    (,ρ')≥ 1-4C_0C_1C_2Λ ^4· 2^6Λ.

By    choosing C=4C_0C_1C_2, we deduce the claim.


We are left to prove Propositions <ref>, <ref> and 
 <ref>. This is done in a reverse manner.




   Note that  participates in most of  the checks in  regardless of the specific challenge. 
   In clause:Check_1_in_associated_test, it appears 3+2|S_xy| times, and since |S_xy|≤ 2Λ, this is bounded by 3+4Λ. 
   In clause:Check_2_in_associated_test it is not used. 
   In clause:Check_3_in_associated_test it is used at most |S_xy| times, which is bounded from above  by 2Λ. 
   Lastly, the number of times   appears in L_xy(γ) is bounded by the number of subsets of S_xy∪{} that contain it, which is 2^|S_xy|≤ 2^2Λ.   All in all, the significance of  is at most (3+4Λ)+2Λ+2^2Λ≤ 4· 2^2Λ. 
    
    Now, a variable  in S_x appears only if we sampled an edge in E with x as one of its endpoints. If this happened, it will appear twice in clause:Check_1_in_associated_test, 2+2|S_x| times in clause:Check_2_in_associated_test, at most twice in clause:Check_3_in_associated_test, and at most 2^|S_xy∪{}|-1 times in clause:Check_4_in_associated_test. Again, |S_x|≤Λ and |S_xy|≤ 2Λ, and thus  appears at most 2+2+2Λ+2+2^2Λ≤ 3· 2^2Λ times when an edge containing x is sampled. This means that the significance of  is at most μ(x)· 3· 2^2Λ. Choosing C_2=4 completes the proof.




Let σS̃→(2n) be a finitely described strategy that passes clause:Check_1_in_associated_test, clause:Check_2_in_associated_test and clause:Check_3_in_associated_test of  with probability 1. Then, 
from perfectly passing clause:Check_1_in_associated_test and  clause:Check_2_in_associated_test, σ is a permutation strategy, and by perfectly passing clause:Check_3_in_associated_test  it is Z-aligned. The other direction was proved in the completeness analysis in Section <ref>. We are going to prove that 

    (,ρ)≥ 1- 2· 2^2Λ(1-(,σ)),
 and thus C_1=2 satisfies the claim.

We start by giving expressions for the success probability of σ in , and of the associated ρ in 𝒢 respectively. 
Fix an edge xy∈ E, and denote by σ_x=σ|_S_x∪{} and σ_y=σ|_S_y∪{} the appropriate restrictions of the permutation strategy σ.  The intersection of orbits O_x∩ O_y, where O_x is an orbit of σ_x and O_y is an orbit of σ_y, partitions [2n] into disjoint sets. Fixing a pair of orbits O_x,O_y such that O_x∩ O_y≠∅,  by Claim <ref>, there is a set L_xy^O_x,O_y  of subsets of  S_xy∪{} such that 

    ∀⋆∈ O_x∩ O_y   L^O_x,O_y_xy=L_xy(⋆)=L̃_xy((σ,⋆)).
 
For α∈𝔽_2^S_xy∪{}, define

    ^α=∏_∈ S_x∪{}^α()   and    ^α=∏_∈ S_y^α().

Let Υ̃^O_x,O_y(α) be the set of all vertices ⋆∈ O_x∩ O_y such that σ(^α).⋆≠σ(^α).⋆, namely all vertices in the intersection of orbits that do not satisfy the subgroup test constraint induced by α.  Let 

    Υ̃^O_x,O_y_xy = ⋃_α∈ L^O_x,O_y_xyΥ̃^O_x,O_y(α),

and let 

    Υ̃_xy=⋃_O_x,O_yΥ̃^O_x,O_y_xy,

where the union is over all orbits of σ_x and σ_y that have a non-empty intersection.
Then Υ̃_xy is the set of all  ⋆∈ [2n] such that D̃_xy( (σ,⋆))=0, hence

    (,σ)=1- _xy∼μ|Υ̃_xy|/2n =1-_xy∼μ_⋆∈ [2n]_xy(⋆)   ,

 where
 
    _xy(⋆)=|Υ̃_xy^O_x,O_y|/|O_x∩ O_y|

 and O_x and O_y are the unique σ_x and σ_y orbits for which ⋆∈ O_x∩ O_y.
 
Recall that according to the procedural sampling described in  Claim <ref>, any pair v⃗∈ B_x^-, u⃗∈ B^-_y of Fourier basis elements supported on σ_x-orbit O_x and σ_y-orbit O_y respectively, is sampled with probability 2⟨v⃗|u⃗⟩^2/|O_x∩ O_y| (conditional on the sampled ⋆ being in O_x∩ O_y), and it induces an “assignment” γ S_xy∪{}→_2 for which γ()=1.
 Let Υ^O_x,O_y(α) be the set of all such (v⃗,u⃗) for which 
  γ does not satisfy a specific linear constraint α∈_2^S_xy∪{}, namely ∑_∈ S_xy∪{}γ()α()≠ 0. 
  Let  
 
    Υ^O_x,O_y_xy = ⋃_α∈ L^O_x,O_y_xyΥ^O_x,O_y(α)

 be the collection of Fourier basis pairs (v⃗,u⃗)∈ B_x^-× B_y^- supported on O_x,O_y, such that the assignment γ induced by the pair is rejected by D_xy. 
 Finally let 
 
    Υ_xy=⋃_O_x,O_yΥ^O_x,O_y_xy⊆ B^-_x× B^-_y,

 which is exactly the collection of possible sampled pairs by the quantum strategy ρ induced by σ, given that the edge xy was asked,  that produce an assignment γ satisfying D_xy(γ)=0. 
 Hence,

    (,ρ)=1-_xy∼μ∑_(v⃗,u⃗)∈Υ_xy⟨v⃗|u⃗⟩^2/n= 1-_xy∼μ_⋆∈ [2n]δ_xy(⋆) ,

where 

    δ_xy(⋆) =∑_(v⃗,u⃗)∈Υ^O_x,O_y_xy2⟨v⃗|u⃗⟩ ^2/|O_x∩ O_y|

and O_x,O_y are the unique orbits for which ⋆∈ O_x∩ O_y.
Observe the following claim.

    For every α∈𝔽_2^S_xy∪{},  σ_x-orbit O_x and σ_y-orbit O_y such that O_x∩ O_y≠∅, we have

    ∑_(v⃗,u⃗)∈Υ^O_x,O_y(α)⟨v⃗|u⃗⟩ ^2 ≤ 1/2|Υ̃^O_x,O_y(α)|  .


Let us show how to deduce our conclusion assuming the  validity of this claim, and then we will address the claim itself.
Given ⋆∈ O_x∩ O_y, we have

    δ_xy(⋆)     (<ref>)=∑_(v⃗,u⃗)∈Υ^O_x,O_y_xy2⟨v⃗|u⃗⟩ ^2/|O_x∩ O_y|
          ≤ ∑_α∈ L_xy^O_x,O_y ∑_(v⃗,u⃗)∈Υ^O_x,O_y(α)2⟨v⃗|u⃗⟩ ^2/|O_x∩ O_y|
        Claim <ref>≤∑_α∈ L_xy^O_x,O_y |Υ̃^O_x,O_y(α)|/|O_x∩ O_y| .

Now, for every α∈ L_xy^O_x,O_y,  we have by (<ref>) that Υ̃^O_x,O_y(α)⊆Υ̃^O_x,O_y_xy, which using the above analysis means 

    δ_xy(⋆)≤∑_α∈ L_xy^O_x,O_y |Υ̃^O_x,O_y_xy|/|O_x∩ O_y|≤ 2^2Λ+1·_xy(⋆),

where the last inequality uses (<ref>) and  the fact |L^O_x,O_y_xy|≤|_2^S_xy∪{}|≤  2^2Λ+1. 
Therefore,

    1-(𝒢,ρ)(<ref>)=_xy∼μ_⋆∈ [2n][δ_xy(⋆)]≤ 2· 2^2Λ_xy∼μ_⋆∈ [2n][_xy(⋆)](<ref>)=2· 2^2Λ(1-( 𝒯̃,σ)),

proving (<ref>) and thus the proposition. 




Let Γ̃^O_x,O_y(α) be the complement of Υ^O_x,O_y(α) in O_x∩ O_y, namely 

    Γ̃^O_x,O_y(α)= (O_x∩ O_y) ∖Υ̃^O_x,O_y(α).

Recall the notation ^α and ^α from (<ref>).
For ⋆∈ O_x∩ O_y, we have σ(^α).⋆∈ O_x and σ(^α).⋆∈ O_y by the fact O_x is a σ_x-orbit and O_y is a σ_y-orbit. If ⋆∈Γ̃^O_x,O_y(α), then by its definition we have ♢:=σ(^α).⋆=σ(^α).⋆, and ♢∈ O_x∩ O_y.  As σ(^α) and σ(^α) are involutions, we have σ(^α).♢=⋆=σ(^α).♢, which implies ♢∈Γ̃^O_x,O_y(α).  All in all, Γ̃^O_x,O_y(α) is invariant under the actions of ^α and ^α — graphically, if we add ^α and ^α labels to the Schrier graph of the action σ on [2n], then they agree on every ⋆∈Γ̃^O_x,O_y(α) and are contained in it.

For (v⃗,u⃗) ∈Υ^O_x,O_y(α)  we know that  σ(^α)v⃗ =±v⃗ while σ(^α)u⃗=∓u⃗. 
This means that for every ^α-labeled edge between vertices in Γ̃^O_x,O_y(α), v⃗ and u⃗ agree on one of its endpoints and evaluate to opposites on the other endpoint. 
This implies

    ⟨v⃗|u⃗⟩   =∑_⋆∈ O_x∩ O_yv⃗(⋆)u⃗(⋆)
       =∑_⋆∈Υ̃^O_x,O_y (α)v⃗(⋆)u⃗(⋆)+∑_⋆∈Γ̃^O_x,O_y(α)v⃗(⋆)u⃗(⋆)
       =∑_⋆∈Υ̃^O_x,O_y (α)v⃗(⋆)u⃗(⋆)+1/2∑_⋆∈Γ̃^O_x,O_y(α)v⃗(⋆)u⃗(⋆)+v⃗(σ(^α).⋆)u⃗(σ(^α).⋆)_=0
       =⟨v⃗· 1_Υ̃^O_x,O_y (α)|u⃗⟩,

where  1_Υ̃^O_x,O_y is the indicator function of the set Υ̃^O_x,O_y, and · is the pointwise product of functions.
Thus,

    ∑_(v⃗,u⃗)∈Υ^O_x,O_y(α)⟨v⃗|u⃗⟩^2   =∑_v⃗Supp(v⃗)=O_x(∑_u⃗
      (v⃗,u⃗)∈Υ^O_x,O_y(α)⟨v⃗· 1_Υ̃^O_x,O_y (α) |u⃗⟩^2)
       ≤∑_v⃗Supp(v⃗)=O_x(∑_u⃗Supp(u⃗)=O_y⟨v⃗· 1_Υ̃^O_x,O_y (α) |u⃗⟩^2)
       =(*),

and since 1_Υ̃^O_x,O_y(α) is supported on O_y, we have 

    ∑_u⃗Supp(u⃗)=O_y⟨v⃗· 1_Υ̃^O_x,O_y(α) |u⃗⟩^2=v⃗· 1_Υ̃^O_x,O_y (α)^2=|Υ̃^O_x,O_y (α)|/|O_x| .

Since there are |O_x|/2 many v⃗∈ B^-_x which are supported on O_x, we deduce that
(*)≤|Υ̃^O_x,O_y(α)|/2,
as desired.









































































































A short detour into (quite elementary) group stability results is needed for proving  Proposition <ref>.


 §.§ Interlude — Group stability


 [Almost involutions are close to involutions]
 Let ζ∈(X). Then,
there is τ∈(X) such that τ^2= and
d_H(ζ,τ)= d_H(ζ^2,). 


Let W={ x∈ X |ζ^2 .x=x}.
Note that the restriction ζ|_W is an involution W→ W.
Define 

    τ(x)=ζ(x)     x∈ W,
    
    x     x∉ W.

Then τ^2= and d_H(ζ,τ)=1-|W|/|X|=d_H(ζ^2,).








[Almost fixed point free are close to fixed point free]
        Let ζ∈(X) be an involution. Assume d_H(ζ,)≥ 1-. Then, there is an involution with no fixed points τ∈(Y), where |Y|=2·⌈|X|/2⌉, such that d_H(ζ,τ)≤ 2.


    Let W={x∈ X|ζ.x=x}. Since d_H(ζ,)=1-|W|/|X|, we can deduce that |W|≤|X|. If W is odd, add a vertex to it and make it even. Now that W is even sized, we can choose any perfect matching on it and define τ to be the involution induced by this perfect matching. Extend τ to the rest of X by letting it act as ζ on the vertices out of W. Then the resulting τ is an involution with no fixed points, and 
    
    d_H(ζ,τ)≤|W|+1/|X|≤ 2.



[Almost commuting involutions are close to commuting involutions]
    Given two involutions ζ,τ∈(X), there is an involution τ'∈(X) such that
    
    [τ',ζ]=   and   d_H(τ,τ')≤ d_H([τ,ζ],).

    
    
   Let δ=d_H([τ,ζ],), and let W={x∈ X |ζτ .x=τζ .x}. Then |W|= (1-δ)|X|. Moreover, for every x
    ∈ W, 
    
    ζτ.(ζ. x)   =ζ .(τζ .x)=ζ^2 τ .x=τ .x=τζ .(ζ .x)ζ .x∈ W;
    τζ. (τ .x)   =τ .(ζτ .x)= τ^2 ζ .x=ζ.x=ζτ .(τ .x)τ .x∈ W.

    Hence, W is invariant under the actions of ζ and τ. Thus, the following is well defined
    
    τ'(x)=τ(x)     x∈ W,
    
                x     x∉ W.

    We therefore can conclude that d_H(τ,τ')≤ 1-|W|/|X|=δ, and 
    
    ∀ x∈ W   τ'ζ.x_∈ W   =τζ. x=ζτ .x=ζτ'. x;
    ∀ x∉ W   τ'ζ. x_∉ W   = ζ .x=ζτ'.x,

    which implies [τ',ζ]= as required.
 
 
     Claim <ref> can be proved even without assuming that the permutations are involutions (while allowing both permutations to change and not only τ). This was originally proved in <cit.> and was later reproved with effective bounds in <cit.>.
 


The following is a special case of item 1 of Theorem 2 in <cit.>.
[Almost actions of finite groups are close to actual actions]
    Let G be a finite group, and let f G→(X) be a function. Assume that for every g,h∈ G, d_H(f(gh),f(g)f(h))≤. Then, there is a homomorphism φ G→(X) such that 
    
    ∀ g∈ G   d_H(f(g),φ(g))≤ |G|^2.



    Let x∈ X be a vertex such that 
    
    ∀ g,h∈ G   f(gh).x=f(g)f(h).x.

    Then, if O⊆ X is the orbit of x with respect to the action of (the group generated by) (f), then for every y∈ O we have f(gh).y=f(g)f(h).y. To prove that, it is enough to prove it for a single step, namely for  y=f(k).x for some k∈ G (the rest is by induction). So, for every g,h∈ G, we have
    
    f(gh).y=f(gh)f(k).x=f(ghk).x=f(g)f(hk).x=f(g)f(h)f(k).x=f(g)f(h).y,

    where all equalities are either f(k).x=y or (<ref>). This means that the restriction of f to its action on O induces a homomorphism from G to (O).
    
    We assumed for every g,h∈ G, that d_H(f(gh),f(g)f(h))≤. Therefore, there are at most |G|^2 many vertices z in X for which there is some g,h∈ G such that f(gh).z≠ f(g)f(h).z. So, the union of orbits as O consists of at least (1-|G|^2)|X| of the vertices in X. So, if we define φ(g).x=f(g).x whenever x satisfies (<ref>), and φ(g).x=x otherwise, then φ is an action of G it satisfies for every g∈ G, d_H(f(g),φ(g))≤ |G|^2.

 






 §.§ Proof of Proposition <ref>

Throughout this proof, we use the term triangle inequality for iterative applications of the Hamming distance's triangle inequality,  as was used  in the proof of Claim <ref>.


   Let σS̃→(n) be a finitely described strategy  for , with (,σ)≥ 1-. By clause:Check_1_in_associated_test, d_H(σ()^2,)≤. By Claim <ref>, there is a function σ_1S̃→(n) that agrees with σ on all ∈ S, but σ_1() is an involution and d_H(σ(),σ_1())≤. Again by clause:Check_1_in_associated_test, d_H(σ(),)≥ 1-, and by the triangle inequality d_H(σ_1(),)≥ 1-2. Let m=⌈n/2⌉. Then, by Claim <ref>, there is a function σ_2S̃→(2m) such that for every ⋆∈ [n]⊆ [2m], we have σ_2().⋆=σ_1().⋆=σ().⋆, and σ_2() is a fixed point free involution with d_H(σ_1(),σ_2())≤ 4. All in all, for every ∈ S∪{} we have d_H(σ_2(),σ())≤ 5.

   Now, let _x be the losing probability when an edge containing x is sampled in . Then, _x∼μ[_x]=. Furthermore, by clause:Check_2_in_associated_test, for every ∈ S_x we have d_H(σ()^2,)≤_x. Therefore, by the triangle inequality, d_H(σ_2()^2,)≤ d_H(σ_2()^2,σ()^2)+d_H(σ()^2,)≤ 10+_x. By Claim <ref>, there is a function σ_3S̃→(2m)  such that  σ_3()=σ_2(), and for every x∈ V and ∈ S_x both σ_3()^2= and d_H(σ_3(),σ_2())≤ 10+_x. Now, by clause:Check_1_in_associated_test, for every ∈ S_x we have d_H(σ()σ(),σ()σ())≤_x. Thus, by the triangle inequality, d_H(σ_3()σ_3(),σ_3()σ_3())≤ 30+2_x. Since σ_3() and σ_3() are involutions, we may apply Claim <ref> and find a function σ_4S̃→(2m) such that σ_4()=σ_3(), σ_4()σ_4()=σ_4()σ_4() and d_H(σ_4(),σ_3())≤ 30+2_x. Again by the triangle inequality, we deduce that d_H(σ(),σ_4())≤ 5, and that for ∈ S_x we have d_H(σ(),σ_4())≤ 40 +3_x.

   Recall that S_x={^x,i| i≤ℓ(x)} (ignoring readability). We extend σ_4 to _2^S_x in the following way: For ξ S_x→_2, let
   
    σ_4(ξ)=∏_i=1^ℓ(x)σ_4()^ξ(),

   where the product is ordered according to the index i. By clause:Check_2_in_associated_test, for every ,'∈ S_x we have 
    d_H(σ()σ('),σ(')σ())≤_x.

   Thus, by the triangle inequality, d_H(σ_4()σ_4('),σ_4(')σ_4())≤ 160+ 13_x. Note also that the images of σ_4 are involutions. Therefore, for every ξ_1,ξ_2 S_x→_2 we have 
   
    d_H(σ_4(ξ_1+ξ_2),σ_4(ξ_1)σ_4(ξ_2))≤ℓ(x)^2(160+ 13_x)≤Λ^2(160+ 13_x).

   By Claim <ref> applied to each _2^S_x individually, we have a function σ_5S̃→(2m) such that σ_5(S_x) commutes and consists only of involutions, and 
    d_H(σ_4(),σ_5())≤ |_2^S_x|^2 ·Λ^2(160+13_x)≤ 2^2Λ·Λ^2(160+13_x).
 Furthermore, because of the way σ_5 is constructed in the proof of Claim <ref>, by letting σ_5()=σ_4(), the image of  is still a central involution with no fixed points. So, σ_5 always passes clause:Check_1_in_associated_test and clause:Check_2_in_associated_test. 

   Since σ passes clause:Check_3_in_associated_test with probability _x when an edge with x∈ V  is sampled, we deduce that every readable variable ∈ S_x^ satisfies

    _⋆∈ [n][σ().⋆=⋆ or σ().⋆=σ().⋆]≥ 1-_x.

Hence, by the triangle inequality, 
   
    _⋆∈ [2m][σ_5().⋆=⋆ or σ_5().⋆=σ_5().⋆]   ≥ 1-_x-d_H(σ_5(),σ())-d_H(σ_5(),σ())
       ≥ 1-_x-5-2^2Λ·Λ^2(160+13_x)
       ≥ 1-2^2Λ·Λ^2(165+14_x).

   By the fact that σ_5(S_x) commutes for every x∈ V,  the property σ_5().⋆=⋆ implies σ_5().♢=♢ for every ♢∈ O_x, where O_x is the orbit of ⋆ induced by σ_5(S_x) (which we usually denoted as (σ_5)_x beforehand). Since σ_5() commutes with all images, the same is true for the property σ_5().⋆=σ_5().⋆.  
   Hence, in each orbit of σ_5(S_x), either all vertices ⋆ satisfy σ_5().⋆=⋆ or σ_5().⋆=σ_5().⋆ or none of them. 
   We can thus define a function σ_6 that agrees with σ_5 on the orbits in which all readable variables ∈ S_x^ act appropriately, and on the other orbits we just define for every ∈ S_x that σ_6().⋆=⋆ (note that we keep  acting the same no matter what). So, we have 
   
    ∀ x∈ V, ∈ S_x    d_H(σ_6(),σ_5())   ≤ |S_x^|· 2^2Λ·Λ^2(165+14_x)
       ≤ 2^2Λ·Λ^3(165+14_x),

and now σ_6 always passes clause:Check_1_in_associated_test, clause:Check_2_in_associated_test and clause:Check_3_in_associated_test. By the triangle inequality, d_H(σ_6(),σ())≤ 5 and for every ∈ S_x we have

    d_H(σ_6(),σ())   ≤ 2^2Λ·Λ^3(165+14_x)+2^2Λ·Λ^2(160+13_x)
       +30+2_x+10+_x+5
       ≤ 2^2ΛΛ ^3(370+30_x).

Choosing σ'=σ_6 and C_0=370, we deduce the claim.



    How better can the parameters in Propositions <ref>, <ref> and <ref> be? We know how to  make the parameters in Proposition <ref>  polynomial in Λ by implementing a Hadmamard code in every vertex x∈ V and using <cit.> or <cit.> in the analysis instead of <cit.> — since we did not see any theoretical gain by doing that, and it was a lengthier proof, we decided not to implement this parameter improvement. We are also not sure whether the dependence in Proposition <ref> is needed or not. Finally, the significance analysis performed in Section <ref> is very crude. Can a better analysis allow to remove the dependence on Λ altogether, at least for some rich enough family of games?




    



§ SUBGROUP STRATEGIES

We need to decide if and where to keep this

    A subgroup strategy for a non-local game  consists of an IRS κ∈((  S∪{})) with the following properties for κ-a.e. subgroup H.

    
  * The image of  is a central involution with no fixed points. Namely, we have 
    
    ^2 ∈ H, ∉ H,
       ∀∈ S   [,]∈ H.

    
  * The rest of the images are involutions as well. Namely,
    
    ∀∈ S   ^2 ∈ H.

    
  * The image of S_x commutes. Namely, for every fixed vertex x∈ V, and 
    
    ∀,'∈ S_x   [,'] ∈ H.





 Let κ∈((  S∪{})) be a subgroup strategy for .
The quantum strategy ρ induced by κ is defined as follows. Let =(S∪{}) and sub() be the space of pairs (H,Hg) with H ∈sub() and Hg ∈ H \. Let κ̃ be the measure on this space which disintegrates as 

    dκ̃(H,Hg) = dκ(H).

In other words, we are putting counting measure on the fibers of the map sub() →sub(). So if F is a non-negative Borel function on sub() then

    ∫ F(H,Hg) dκ̃(H,Hg) = ∫∑_Hg F(H,Hg)  dκ(H).

Let L^2(κ̃) denote the associated L^2-space. 

For f ∈, define an operator σ̃(f) on L^2(κ̃) by

    (σ̃(f)·ϕ)(H,Hg) = ϕ(f^-1Hf, f^-1Hg) =ϕ (f^-1Hf, f^-1Hf · f^-1g).

It is an exercise to verify that σ̃ defines a homomorphism from  into the unitary group of L^2(κ̃). 

Let L^2_1(,κ̃) ⊂ L^2(κ̃) be the subspace of functions ϕ such that σ̃()·ϕ = -ϕ. Let 

    ^_1: L^2(κ̃) → L^2_1(,κ̃)

be the orthogonal projection map onto L^2_1(,κ̃). Let

    ^_1: L^2(κ̃) → L^2(κ̃)

be the composition of ^_1 with the inclusion map L^2_1(,κ̃) → L^2(κ̃).

Note that (^_1)^*^_1=^_1 and ^_1(^_1)^*=_n.

Given a Hilbert space ℋ, let B(ℋ) denote the algebra of bounded operators on ℋ. Define ρ S→ B( L^2_1(,κ̃)) by
      
    ρ()=^_1 σ̃()(^_1)^*.








    Let  κ be a subgroup strategy. Then, the quantum strategy ρ induced by κ, as in Definition <ref>, is indeed a quantum strategy for , as in Definition <ref>.


The proof is the exact same as the proof of Claim <ref>. 








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