###### Proof.

###### Directed cycles with zero weight in $\mathbb{Z}_{p}^{k}$

###### Abstract

For a finite abelian group $A$, define $f(A)$ to be the minimum integer such
that for every complete digraph $\Gamma$ on $f$ vertices and every map
$w:E(\Gamma)\rightarrow A$, there exists a directed cycle $C$ in $\Gamma$ such
that $\sum_{e\in E(C)}w(e)=0$. The study of $f(A)$ was initiated by Alon and
Krivelevich (2021). In this article, we prove that
$f(\mathbb{Z}_{p}^{k})=O(pk(\log k)^{2})$, where $p$ is prime, with an
improved bound of $O(k\log k)$ when $p=2$. These bounds are tight up to a
factor which is polylogarithmic in $k$.

##  1 Introduction

Generally speaking, the area of ‘zero-sum’ Ramsey theory concerns the study of
objects weighted by the elements of a group, and when a substructure of total
weight zero can be found. This dates back to 1960 and the celebrated
Erdős–Ginzburg–Ziv [9] theorem which asserts that, for every integer $m\geq
2$, every sequence of $2m-1$ elements in $\mathbb{Z}_{m}$ contains a
subsequence of $m$ elements that sum to $0$. Since then, a wide variety of
interesting variations have been studied; see Caro [6] for a survey of this
topic.

For a finite abelian group $A$, define $f(A)$ to be the minimum integer such
that for every complete digraph $\Gamma$ on $f$ vertices and every map
$w:E(\Gamma)\rightarrow A$, there exists a directed cycle $C$ in $\Gamma$ such
that $\sum_{e\in E(C)}w(e)=0$. We call $w$ a _weighting_ of $E(\Gamma)$ and
$C$ a $w$-_zero-sum cycle_. When it is clear which map $w$ is being referred
to, we will simply call $C$ a _zero-sum cycle_. The question of determining
$f(A)$ arose in a paper by Alon and Krivelevich [3] from 2021, who showed that
for any integer $q$, if $f=f(\mathbb{Z}_{q})$, then every $K_{2f}$-minor
contains a cycle whose length is divisible by $q$ (see the end of Section 3 in
[3]). In fact, they proved more generally that for every subcubic graph $H$
and positive integer $q$, there is a (finite) number $g=g(H,q)$ such that
every $K_{g}$-minor contains a subdivision of $H$ where each edge is replaced
by a path of length which is divisible by $q$. Their bound on $g(H,q)$ was
improved by Das, Draganić, and Steiner [7] who showed that $g(H,q)=O(|H|q)$,
which is tight up to a constant factor.

Alon and Krivelevich [3] proved that $f(\mathbb{Z}_{p})\leq 2p-1$, for $p$
prime, and $f(\mathbb{Z}_{q})=O(q\log q)$, for any integer $q\geq 2$. This
result was improved upon and generalised by Mészáros and Steiner [10], who
showed that $f(A)\leq 8|A|$ for any finite abelian group $A$, and in
particular, that $f(\mathbb{Z}_{p})\leq\frac{3}{2}p$ for prime $p$. Recently,
this was improved upon by Berendsohn, Boyadzhiyska, and Kozma [5], and
independently by Akrami, Chaudhury, Garg, Mehlhorn, and Mehta [1], who gave a
beautifully slick proof to show that $f(B)\leq 2|B|-1$, where $B$ is any
finite (not necessarily abelian) group. In forthcoming work of Campbell,
Hendrey, Gollin, and Steiner, this is improved to a tight bound
$f(\mathbb{Z}_{q})=q+1$ for every positive integer $q$.

Our main result improves upon the results in [10, 5, 1] to give a sublinear
bound in $|A|$ when $A=\mathbb{Z}_{p}^{k}$ and $p$ is a prime.

######  Theorem 1.1.

Let $p$ be a prime and let $k\geq 1$ be an integer. Then

$f(\mathbb{Z}_{p}^{k})\leq 600p\cdot k(\log_{2}(10k))^{2}.$

We obtain a stronger result when $p=2$, which is tight up to an $O(\log k)$
factor.

######  Theorem 1.2.

Let $k\geq 2$ be an integer. Then

$f(\mathbb{Z}_{2}^{k})\leq 600k\log_{2}(2k).$

A simple construction shows that $f(\mathbb{Z}_{p}^{k})\geq(p-1)k$. Indeed,
let $e_{1},\ldots,e_{k}$ be the elementary basis elements of
$\mathbb{Z}_{p}^{k}$. Consider the complete digraph $\Gamma$ on $(p-1)k$
vertices, let $\\{V_{1},\ldots,V_{k}\\}$ be an equipartition of $V(\Gamma)$,
and label a directed edge $xy$ by $e_{i}$ whenever $x\in V_{i}$. It is easy to
see that, with this weighting, there are no zero-sum cycles in $\Gamma$, as
every cycle contains at most $p-1$ edges labelled $e_{i}$, for every
$i\in[k]$. Thus $f(\mathbb{Z}_{p}^{k})\geq(p-1)k$, as claimed. The results in
Theorem 1.1 and Theorem 1.2 are thus tight up to a factor which is
polylogarithmic in $k$.

This problem was independently investigated by Sidorenko and Steiner
(unpublished) who showed $f(\mathbb{Z}_{p}^{k})=O(pk^{2}\log k)$ with an
improved bound of $O(k^{2})$ when $p=2$.

The proofs of Theorems 1.1 and 1.2 follow the same strategy. In Lemma 4.1
below, we show that $f(\mathbb{Z}_{p}^{k})$ can be bounded by an expression
involving the largest size of a ‘reduced’ multisubset of $\mathbb{Z}_{p}^{k}$.
This parameter is bounded in general for $\mathbb{Z}_{p}^{k}$ in Theorem 2.1,
and a better bound is easily obtained when $p=2$, as we will see in
Observation 2.2.

In Section 2 we introduce the notion of reduced sets, make some preliminary
observations and prove Theorem 2.1. Then, in Section 3, we show how we can use
particular subgraphs, called ‘gadgets’, collectively to find zero-sum cycles.
These allow us to deduce information about the weights on our digraph and, in
Section 4, to prove Lemma 4.1. This in turn completes the proofs of Theorem
1.1 and Theorem 1.2. We conclude in Section 5 with some discussion about
bounding the largest size of a ‘reduced’ multisubset of $\mathbb{Z}_{p}^{k}$
and other directions for future research.

###  1.1 Notation and conventions

All logarithms will be taken in base $2$.

For sets $A$ and $B$, we write $A+B:=\\{a+b:a\in A,b\in B\\}$.

We will often treat collections $S$ of elements of $\mathbb{Z}_{p}^{k}$ as
collections of vectors in $\mathbb{F}_{p}^{k}$. As such, we will refer to the
span or dimension of such a collection $S$. We will use 0 to denote the
identity element of any group $A$.

We will often be working with multisets. In order to minimise confusion for
the reader, we now define some notation for multiset operations that will be
used throughout.

Given an abelian group $A$ and a multiset $S$, write
$S\>\widetilde{\subseteq}\>A$ to denote that $S$ is a multiset with the
property that $s\in A$ for every $s\in S$. Write $T\subseteq S$ to denote that
$T$ is a (multi)subset of $S$. We will write $S-T$ to denote the multiset
obtained from $S$ by removing one copy of each element of the multiset $T$.

Define the _multiset union_ of sets $A_{1},\ldots,A_{k}$, denoted by
$\widetilde{\bigcup}_{i\in[k]}A_{i}$ to be the multiset whose elements can be
partitioned into $A_{1},\ldots,A_{k}$. If $a_{1},\ldots,a_{k}$ are elements of
$A$, write $\widetilde{\bigcup}_{i\in[k]}a_{i}$ to denote the multiset
containing $\\{a_{1},\ldots,a_{k}\\}$.

##  2 Reduced sets

Given a multiset $S$ with elements in an abelian group $A$, the _sumset_ of
$S$, denoted $\Sigma(S)$, is defined to be the set of all subset sums of $S$,
namely

$\Sigma(S):=\left\\{\sum_{t\in T}t:T\subseteq S\right\\},$

where the sum of elements in the empty set is defined to be zero, and hence
$|\Sigma(\emptyset)|=1$. In particular, $0\in\Sigma(S)$ for every multiset
$S\>\widetilde{\subseteq}\>A$. Note that, when working over $\mathbb{F}_{2}$,
the sumset of $S$ is equal to the span of $S$. Say that $S$ is _reduced_ if
$|\Sigma(S)|>|\Sigma(S-\\{s\\})|$, for all $s\in S$. Let $h_{p}(k)$ be the
size of a largest reduced multiset in $\mathbb{Z}_{p}^{k}$.

The main result in this section is a general upper bound on $h_{p}(k)$.

######  Theorem 2.1.

$h_{p}(k)\leq(p-1)(\log k+1)\cdot k$.

This shows that the lower bound $h_{p}(k)\geq k(p-1)$, from Observation 2.2
(viii) below, is tight up to a $\log k+1$ factor.

We begin by gathering together some straightforward observations about reduced
multisets.

######  Observation 2.2.

Let $A$ be a finite abelian group and let $S\>\widetilde{\subseteq}\>A$.

  1. (i)

If $S$ is reduced, then the identity element $0\in A$ is not an element of
$S$.

  2. (ii)

There exists $S^{\prime}\subseteq S$ such that $S^{\prime}$ is reduced and
$\Sigma(S)=\Sigma(S^{\prime})$.

  3. (iii)

If $S$ is reduced, then every $T\subseteq S$ is reduced.

  4. (iv)

Suppose that $S$ is reduced and $A=\mathbb{Z}_{p}^{k}$. Let
$f:\mathbb{Z}_{p}^{k}\to\mathbb{Z}_{p}^{k}$ be an injective linear map. Then
$f(S):=\\{f(s):s\in S\\}$ is reduced.

  5. (v)

$h_{p}(k)+h_{p}(\ell)\leq h_{p}(k+\ell)$.

  6. (vi)

$h_{p}(1)=p-1$.

  7. (vii)

A multisubset of $\mathbb{Z}_{2}^{k}$ is reduced if and only if it is linearly
independent. In particular, $h_{2}(k)=k$.

  8. (viii)

$h_{p}(k)\geq k(p-1)$.

###### Proof.

For (i), observe that $\Sigma(S)=\Sigma(S-\\{0\\})$.

For (ii), if $S$ is not reduced then sequentially remove elements $t$ such
that $\Sigma(S-\\{t\\})=\Sigma(S)$ until a reduced multisubset $S^{\prime}$
remains.

For (iii), if $T\subseteq S$ and $\Sigma(T)=\Sigma(T-\\{t\\})$ for some $t\in
T$, then
$\Sigma(S-\\{t\\})=\Sigma(S-T)+\Sigma(T-\\{t\\})=\Sigma(S-T)+\Sigma(T)=\Sigma(S)$,
contradicting the assumption that $S$ is reduced. Thus
$|\Sigma(T)|>|\Sigma(T-\\{t\\})|$ for every $t\in T$, showing that $T$ is
reduced.

Item (iv) follows from the observation that $\Sigma(f(T))=f(\Sigma(T))$ for
every multisubset $T$ of $\mathbb{Z}_{p}^{k}$.

For (v), let $S\>\widetilde{\subseteq}\>\mathbb{Z}_{p}^{k}$ and
$T\>\widetilde{\subseteq}\>\mathbb{Z}_{p}^{\ell}$ be reduced with respective
sizes of $h_{p}(k)$ and $h_{p}(\ell)$. Define $R:=\\{(s,0):s\in
S\\}\cup\\{(0,t):t\in
T\\}\subseteq\mathbb{Z}_{k}^{p}\times\mathbb{Z}_{p}^{\ell}$. It is easy to see
that $R$ is a reduced multiset in
$\mathbb{Z}_{p}^{k}\times\mathbb{Z}_{p}^{\ell}\cong\mathbb{Z}_{p}^{k+\ell}$ of
size $h_{p}(k)+h_{p}(\ell)$, showing $h_{p}(k+\ell)\geq h_{p}(k)+h_{p}(\ell)$.

For (vi), suppose that $S\>\widetilde{\subseteq}\>\mathbb{Z}_{p}$ is reduced.
Enumerate the elements of $S$ as $\\{s_{1},\ldots,s_{k}\\}$. By (iii),
$\\{s_{1},\ldots,s_{i}\\}$ is reduced for all $i\in[k]$, and so

$|\Sigma(S)|=\big{|}\Sigma(\\{s_{1},\ldots,s_{k}\\})\big{|}\geq\big{|}\Sigma(\\{s_{1},\ldots,s_{k-1}\\})\big{|}+1\geq\ldots\geq|\Sigma(\emptyset)|+k=k+1.$

Since, trivially, $|\Sigma(S)|\leq p$, we have $|S|=k\leq p-1$.

Thinking of $\mathbb{Z}_{2}^{k}$ as a vector space over $\mathbb{F}_{2}$, the
sumset of a set $S\subseteq\mathbb{Z}_{2}^{k}$ is the same as the span of $S$.
Item (vii) follows.

Finally, for (viii), notice that the multiset consisting of $p-1$ copies of
each elementary basis element of $\mathbb{Z}_{p}^{k}$ is a reduced multiset of
size $k(p-1)$. ∎

###  2.1 Proof of Theorem 2.1

Before giving the proof, we will state two preliminary results. The first is
the Matroid Packing theorem, due to Edmonds [8]. Before stating it, we will
define matroids and some relevant notions.

A _matroid_ is a pair $(V,\mathcal{I})$, where $V$ is a set and $\mathcal{I}$
is a non-empty family of subsets of $V$, referred to as _independent sets_ ,
which is closed under taking subsets, and satisfies the following
_augmentation property_ : if $I,I^{\prime}\in\mathcal{I}$ satisfy
$|I|>|I^{\prime}|$, then there exists $x\in I\setminus I^{\prime}$ such that
$I^{\prime}\cup\\{x\\}\in\mathcal{I}$. Define the _rank_ of a subset
$S\subseteq V$, denoted $\operatorname{rank}(S)$, as the size of a largest
independent set contained in $S$. A _base_ is an independent set of size
$\operatorname{rank}(V)$.

######  Theorem 2.3 (Matroid Packing theorem, Edmonds [8]).

A matroid $M=(V,\mathcal{I})$ contains $t$ pairwise disjoint bases if and only
if every $T\subseteq V$ satisfies $|V|-|T|\geq
t\cdot(\operatorname{rank}(V)-\operatorname{rank}(T))$.

We draw the following almost immediate conclusion regarding disjoint bases in
multisubsets of vector spaces. Given a (multi)subset $S$ of a vector space,
define $\operatorname{rank}(S):=\dim(\operatorname{span}(S))$.

######  Corollary 2.4.

Let $U$ be a finite dimensional vector space and let
$S\>\widetilde{\subseteq}\>U$ have full rank. Then $S$ contains $t$ pairwise
disjoint bases of $U$ if and only if every multisubset $T\subseteq S$
satisfies $|S|-|T|\geq t\cdot(\operatorname{rank}(S)-\operatorname{rank}(T))$.

###### Proof.

We define a matroid $M=(V,\mathcal{I})$, as follows. For each element $s\in S$
of multiplicity $m_{s}$, add the elements $(s,1),\ldots,(s,m_{s})$ to $V$ and
say that each of these elements _corresponds_ to $s$. So $V$ is a set (rather
than a multiset) which emulates $S$. Define $\mathcal{I}$ to be the subsets of
$V$ with the property that the collection of vectors they correspond to in $S$
is linearly independent. It is easy to check that $M$ is indeed a matroid and
that a base in $M$ corresponds to a basis of $U$. It follows from Theorem 2.3
that $M$ contains $t$ pairwise disjoint bases of $U$ if and only if
$|V|-|T|\geq t\cdot(\operatorname{rank}(V)-\operatorname{rank}(T))$ for every
$T\subseteq V$. This proves the statement, as $|V|=|S|$ and
$\operatorname{rank}(V)=\operatorname{rank}(S)$. ∎

We will also need the following result of Alon, Linial, and Meshulam [4].

######  Theorem 2.5 (Proposition 3.1 in [4]).

Let $S_{1},\ldots,S_{\ell}$ be $\ell$ bases of the vector space
$\mathbb{Z}_{p}^{k}$, where $\ell\geq(p-1)\log k+p-2$, and let
$S=\widetilde{\bigcup}_{i\in\ell}S_{i}$. Then $\Sigma(S)=\mathbb{Z}_{p}^{k}$.

We now have all the pieces we need to prove Theorem 2.1, restated here. See
2.1

###### Proof of Theorem 2.1.

We will prove the statement by induction on $k$. Notice that it follows
directly from Observation 2.2 (vi) if $k=1$, so take $k\geq 2$ and assume that
the statement holds for all $k^{\prime}\in[k-1]$. Write $\ell_{k}=(p-1)(\log
k+1)$. Suppose that $T\>\widetilde{\subseteq}\>\mathbb{Z}_{p}^{k}$ is reduced
and $|T|>\ell_{k}\cdot k$. Let $S\subseteq T$ such that $|S|=\ell_{k}\cdot k$.
By Observation 2.2 (iii), $S$ is reduced. We will show that
$\Sigma(S)=\mathbb{Z}_{p}^{k}$, contradicting the assumption that $T$ is
reduced.

By induction, every $S^{\prime}\>\widetilde{\subseteq}\>S$ with
$r:=\operatorname{rank}(S^{\prime})<k$ satisfies
$|S^{\prime}|\leq\ell_{r}\cdot
r\leq\ell_{k}\cdot\operatorname{rank}(S^{\prime})$, because $S^{\prime}$ is a
reduced multisubset of an $r$-dimensional subspace of $\mathbb{Z}_{p}^{k}$
(using Observation 2.2 (iii) and (iv)). In particular, we have
$\operatorname{rank}(S)=k$, and if $S^{\prime}\>\widetilde{\subseteq}\>S$
satisfies $\operatorname{rank}(S^{\prime})<k$, then

$\frac{|S|-|S^{\prime}|}{\operatorname{rank}(S)-\operatorname{rank}(S^{\prime})}\geq\frac{\ell_{k}\cdot
k-\ell_{k}\cdot\operatorname{rank}(S^{\prime})}{k-\operatorname{rank}(S^{\prime})}\geq\ell_{k}.$

It thus follows from Corollary 2.4 that $S$ is the multiset union of
$\ell_{k}$ pairwise disjoint bases. By Theorem 2.5, this implies that
$\Sigma(S)=\mathbb{Z}_{p}^{k}$, as suffices to complete the result. ∎

##  3 Gadgets

Let $\Gamma$ be a complete digraph, $A$ be an abelian group, and let
$w:E(\Gamma)\rightarrow A$. We will write $uv$ to denote the directed edge
from $u$ to $v$.

######  Definition 3.1 (Gadgets).

Let $u$ and $v$ be distinct vertices in $\Gamma$. A _gadget_ rooted at $(u,v)$
is defined to be a pair $g$ of paths $P$ and $Q$ directed from $u$ to $v$. Let
$u(g),v(g),P(g)$ and $Q(g)$ refer to $u,v,P$ and $Q$, respectively.

Define the _vertex set_ of $g$ by $V(g):=V(P)\cup V(Q)$, and its _value_ by
$g^{*}:=w(Q)-w(P)$, where the _weight_ of a subdigraph $H\subseteq\Gamma$ is
$w(H):=\sum_{e\in E(H)}w(H)$.

The next lemma provides a simple condition for a collection of gadgets to
yield a zero-sum cycle in $\Gamma$. The point is that if the collection of
values is ‘rich’ enough, then by passing through the gadgets, choosing either
$P$ or $Q$ at each one, we are able to use the collection to generate a zero-
sum cycle.

######  Lemma 3.2.

Let $\Gamma$ be a complete digraph whose edges have weights in the abelian
group $A$. Let $\mathcal{G}$ be a family of pairwise vertex-disjoint gadgets
in $\Gamma$. Let $S:=\widetilde{\bigcup}_{g\in\mathcal{G}}g^{*}$. If
$\Sigma(S)=A$, then $\Gamma$ contains a zero-sum cycle.

###### Proof.

Let $\mathcal{G}=\\{g_{1},\ldots,g_{t}\\}$, where $g_{i}=g_{i}(u_{i},v_{i})$.
For ease of notation, define $u_{t+1}:=u_{1}$. Let $I\subseteq[t]$ be a set of
indices such that the following holds, where
$(u_{i},v_{i},P_{i},Q_{i}):=(u(g_{i}),v(g_{i}),P(g_{i}),Q(g_{i}))$.

$\sum_{i\in
I}g^{*}_{i}=-\sum_{i=1}^{t}w(P_{i})-\sum_{i=1}^{t}w(v_{i}u_{i+1}),$

Such a set exists as $\Sigma(S)=A$. Using $\sum_{i\in I}g_{i}^{*}=\sum_{i\in
I}\big{(}w(Q_{i})-w(P_{i})\big{)}$ and rearranging yields

$0=\sum_{i\in I}w(Q_{i})+\sum_{i\notin
I}w(P_{i})+\sum_{i=1}^{t}w(v_{i}u_{i+1}).$ (1)

Now let $C$ be the directed cycle $R_{1}v_{1}u_{2}R_{2}v_{2}u_{2}\ldots
R_{t}v_{t}u_{1}$, where

$R_{i}:=\begin{cases}Q_{i}&i\in I\\\ P_{i}&i\notin I.\end{cases}$

Note that $\sum_{e\in C}w(e)$ is precisely the right hand side of the
expression (1), and so $C$ is a zero-sum cycle. ∎

The following lemma allows us to modify a weighting to ensure all out-edges
from a particular vertex receive weight 0, whilst preserving properties of the
weighting. The idea of locally modifying weights of edges adjacent to a fixed
vertex without creating or destroying zero-sum cycles was also used by
Mészáros and Steiner [10].

######  Lemma 3.3.

Let $\Gamma$ be a complete digraph, let $A$ be an abelian group, and let
$w:E(\Gamma)\to A$. Suppose that there are no zero-sum cycles with respect to
$w$, and let $v_{0}\in V(\Gamma)$. Then there exists $w^{\prime}:E(\Gamma)\to
A$ such that: $w^{\prime}(v_{0}u)=0$ for every $u\in V(\Gamma)$; there are no
zero-sum cycles with respect to $w^{\prime}$; and every gadget $g$ in $\Gamma$
has the same value with respect to both $w$ and $w^{\prime}$.

###### Proof.

Let $u\in V(\Gamma)$ and let $\alpha\in A$. Define
$f_{u,\alpha}(w):E(\Gamma)\to A$ as follows

$f_{u,\alpha}(w)(e):=\left\\{\begin{array}[]{ll}w(e)&\text{ if
$e\in\Gamma\setminus\\{u\\}$}\\\ w(e)+\alpha&\text{ if $e=uv$ for some $v\in
V(\Gamma)$}\\\ w(e)-\alpha&\text{ if $e=vu$ for some $v\in
V(\Gamma)$.}\end{array}\right.$

It is easy to check that $w(C)=f_{u,\alpha}(w)(C)$ for every directed cycle
$C$. Similarly, if $P$ and $Q$ are two directed paths with the same start and
end points, then $w(P)-w(Q)=f_{u,\alpha}(w)(P)-f_{u,\alpha}(w)(Q)$. It follows
that there are no $w$-zero-sum cycles if and only if there are no
$f_{u,\alpha}(w)$-zero-sum cycles, and that the value of any gadget $g$ is the
same with respect to $w$ and $f_{u,\alpha}(w)$.

Let $u_{1},\ldots,u_{n-1}$ be an arbitrary ordering of
$V(\Gamma)\setminus\\{v_{0}\\}$, and let $\alpha_{i}:=w(v_{0}u_{i})$. Let

$w^{\prime}:=f_{u_{n-1},\alpha_{n-1}}\circ\ldots\circ
f_{u_{1},\alpha_{1}}(w).$

Notice that $w^{\prime}(v_{0}u_{i})=w(v_{0}u_{i})-\alpha_{i}=0$, for every
$i\in[n-1]$; equivalently, $w^{\prime}(v_{0}u)=0$ for every $u\in
V(\Gamma)\setminus\\{v_{0}\\}$. By the above discussion, there are no
$w^{\prime}$-zero-sum cycles, and every gadget in $\Gamma$ has the same value
with respect to $w^{\prime}$ and $w$. ∎

We now introduce some notation and terminology that we will use to refer to
particular families of gadgets.

######  Definition 3.4.

Let $\Gamma$ be a complete digraph, let $p$ be prime, let $k$ be a positive
integer and let $w:E(\Gamma)\rightarrow\mathbb{Z}_{p}^{k}$. A collection of
families of gadgets $\mathcal{G}_{1},\ldots,\mathcal{G}_{t}$ in $\Gamma$ is
said to be _useful_ if it satisfies the following properties:

  1. (P1)

The collections $\mathcal{G}_{1},\ldots,\mathcal{G}_{t}$ are pairwise
disjoint, and $V(g_{1})\cap V(g_{2})=\emptyset$ for all distinct
$g_{1},g_{2}\in\bigcup_{i=1}^{t}\mathcal{G}_{i}$;

  2. (P2)

$|V(g)|=3$ for all $g\in\bigcup_{i=1}^{t}\mathcal{G}_{i}$;

  3. (P3)

defining the multiset $U_{i}:=\widetilde{\bigcup}_{g\in\mathcal{G}_{i}}g^{*}$,
the sequence $|\Sigma(U_{1})|,\ldots,|\Sigma(U_{t})|$ is as late as possible
in the lexicographic ordering111Recall that
$a_{1},a_{2},\ldots,a_{t}<b_{1},b_{2},\ldots,b_{t}$ in the _lexicographic
ordering_ on $\mathbb{R}^{t}$ if there is some $i\in[t]$ such that
$a_{i}<b_{i}$ and $a_{j}=b_{j}$ for all $j<i$. on $\mathbb{R}^{t}$ among
sequences satisfying (P1) and (P2).

We say that a useful family is also _reduced_ if the following holds.

  1. (P4)

$U_{i}$ is a reduced multiset, for every $i\in[t]$.

By Observation 2.2 (ii), for every useful family of gadgets
$\mathcal{G}_{1},\ldots,\mathcal{G}_{t}$ in $\Gamma$ there is a useful and
reduced family $\mathcal{G}_{1}^{\prime},\ldots,\mathcal{G}_{t}^{\prime}$ with
$\Sigma(U_{i}^{\prime})=\Sigma(U_{i})$ (where
$U_{i}^{\prime}:=\widetilde{\bigcup}_{g\in\mathcal{G}_{i}^{\prime}}g^{*}$).

For each $i\in[t]$ define

$V_{i}:=\bigcup_{g\in\mathcal{G}_{i}}V(g),\qquad
B_{i}:=\\{x\in\mathbb{Z}_{p}^{k}:x+\Sigma(U_{i})=\Sigma(U_{i})\\}.$ (2)

Observe that the sets $V_{i}$ are pairwise disjoint, that $B_{i}$ is a
subgroup of $\mathbb{Z}_{p}^{k}$, and $B_{i}\subseteq\Sigma(U_{i})$ (using
$0\in\Sigma(U_{i})$). Let $d_{i}:=\dim(B_{i})$, where $B_{i}$ is treated as a
vector subspace of the vector space $\mathbb{Z}_{p}^{k}$.

The next lemma shows that once we have ‘pulled out’ a useful collection of
gadgets, we have control over the weights of many of the edges in the digraph.

######  Lemma 3.5.

Let $\Gamma$ be a complete digraph, let $p$ be prime and let
$w:E(\Gamma)\rightarrow\mathbb{Z}_{p}^{k}$ be a labelling with no zero-sum
cycles. Let $\mathcal{G}_{1},\ldots,\mathcal{G}_{t}$ be a useful collection of
families of gadgets in $\Gamma$, and define $V_{i},B_{i},d_{i}$ as above.
Suppose that $v_{0}\in V(\Gamma)\setminus\bigcup_{j=1}^{t}V_{j}$ satisfies
$w(v_{0}u)=0$ for every vertex $u\neq v_{0}$. Then:

  1. (i)

For each $i\in[t]$, every edge
$e\in\Gamma\setminus(\\{v_{0}\\}\cup\bigcup_{j=1}^{i}V_{j})$ satisfies
$w(e)\in B_{i}$.

  2. (ii)

We have $k>d_{1}>\ldots>d_{t}$.

###### Proof.

Suppose that $e=uv$ violates (i) for some $i\in[t]$. Then the gadget $g$ with
$V(g)=\\{u,v,v_{0}\\}$, $P(g)=v_{0}v$ and $Q(g)=v_{0}uv$ has value

$w(Q(g))-w(P(g))=w(v_{0}uv)-w(v_{0}v)=w(uv),$

which is not in $B_{i}$. By definition of $B_{i}$, this shows
$|\Sigma(U_{i}\cup\\{g^{*}\\})|>|\Sigma(U_{i})|$. Thus, taking
$\mathcal{G}_{i}^{\prime}:=\mathcal{G}_{i}\cup\\{g\\}$, we reach a
contradiction to property (P3).

Define for convenience $B_{0}=\mathbb{Z}_{p}^{k}$ and $d_{0}=k$, and let
$i\in[t-1]$. For (ii), by (i) we have $U_{i}\subseteq B_{i-1}$ (this holds
trivially for $i=1$). It follows that $B_{i}\subseteq\Sigma(U_{i})\subseteq
B_{i-1}$ (where the first inclusion holds by definition of $B_{i}$ and the
second holds because $B_{i-1}$ is a group), and thus $d_{i}\leq d_{i-1}$.
Suppose, towards a contradiction, that $d_{i}=d_{i-1}$. Then $B_{i-1}=B_{i}$
and, by $B_{i}\subseteq\Sigma(U_{i})\subseteq B_{i-1}$, we get
$B_{i}=\Sigma(U_{i})$. But now applying Lemma 3.2 with the subgraph
$\Gamma[V_{i}]$, whose edges are weighted by elements of $\Sigma(U_{i})$,
yields a zero-sum cycle. This contradicts our assumption that $\Gamma$
contains no zero-sum cycles. Hence $d_{i-1}>d_{i}$, proving (ii). ∎

##  4 Zero-sum cycles and reduced sets

Given a prime $p$ and integer $k$, recall that $f(\mathbb{Z}_{p}^{k})$ is the
minimum $f$ such that every complete digraph on $f$ vertices, whose edges have
weights in $\mathbb{Z}_{p}^{k}$, has a zero-sum cycle. For ease of notation,
write $f_{p}(k):=f(\mathbb{Z}_{p}^{k})$. Recall that $h_{p}(k)$ is the maximum
size of a reduced multiset in $\mathbb{Z}_{p}^{k}$.

The main result of this section is the following lemma, which together with
Theorem 2.1 yields Theorem 1.1.

######  Lemma 4.1.

Let $p$ be a prime and let $k\geq 1$ be integer. Then

$f_{p}(k)\leq 60\log(2k)\cdot h_{p}(10k).$ (3)

Before proving Lemma 4.1, we show how it implies Theorem 1.1 and Theorem 1.2.

###### Proof of Theorem 1.1.

By Lemma 4.1 and Theorem 2.1, we have

$\displaystyle f_{p}(k)$ $\displaystyle\leq 60\log(2k)\cdot h_{p}(10k)$
$\displaystyle\leq 60\log(2k)\cdot(p-1)(\log(10k)+1)\cdot(10k)$
$\displaystyle\leq 600p\cdot k(\log(10k))^{2},$

as required. ∎

###### Proof of Theorem 1.2.

By Observation 2.2(vi) and Theorem 2.1, we have

$\displaystyle f_{2}(k)$ $\displaystyle\leq 60\log(2k)\cdot h_{2}(10k)\leq
60\log(2k)\cdot 10k=600\cdot k\log(2k).$

as required. ∎

###### Proof of Lemma 4.1.

We prove the lemma by induction on $k$. Notice that, by [3, 10], $f_{p}(1)\leq
2p$, and by Observation 2.2(vii) we have $60\cdot h_{p}(10)\geq 60\cdot
h_{p}(1)=60(p-1)$, so the statement holds for $k=1$.

Let $\Gamma$ be a complete digraph on $n$ vertices that does not contain a
zero-sum cycle. Our goal is to show that $n\leq 60\log(2k)\cdot h_{p}(10k)$.
Suppose to the contrary that $n\geq 60\log(2k)\cdot h_{p}(10k)$.

Set $t:=\lceil 10\log k\rceil$. Choose
$\mathcal{G}_{1},\ldots,\mathcal{G}_{t}$ to be a collection of families of
gadgets in $\Gamma$ satisfying (P1) and (P2), such that the sequence
$|\Sigma(U_{1})|,\ldots,|\Sigma(U_{t})|$ is as late as possible in the
lexicographic order among all such sequences, where $U_{i}$ is as in (P3), so
that (P3) holds. Notice that such a sequence exists (taking
$\mathcal{G}_{i}=\emptyset$ yields a valid sequence). By replacing $U_{i}$ by
a minimal subset $U_{i}^{\prime}\subseteq U_{i}$ satisfying
$\Sigma(U_{i}^{\prime})=\Sigma(U_{i})$, we may additionally assume that (P4)
holds. Define $V_{i},B_{i},d_{i}$ as in (2).

For each $i\in[t]$, as $U_{i}$ is reduced, $|U_{i}|\leq h_{p}(k)$ and thus
$|V_{i}|\leq 3h_{p}(k)$ (by (P2)), showing $|\bigcup_{i\in[t]}V_{i}|\leq
3t\cdot h_{p}(k)\leq n-1$. Let $v_{0}$ be a vertex not in
$\bigcup_{i\in[t]}V_{i}$. Apply Lemma 3.3 to obtain
$w^{\prime}:E(\Gamma)\to\mathbb{Z}_{p}^{k}$ such that: $w^{\prime}(v_{0}u)=0$
for every vertex $u\neq v_{0}$; there are no $w^{\prime}$-zero-sum cycles; and
every gadget in $\Gamma$ has the same weight with respect to $w$ and
$w^{\prime}$. In particular, a sequence of gadgets is useful with respect to
$w$ if and only if it is useful with respect to $w^{\prime}$, and similarly
for useful and reduced sequences. Altogether, this means that we may and will
replace $w$ by $w^{\prime}$, allowing us to assume $w(v_{0}u)=0$ for $u\in
V(\Gamma)\setminus\\{v_{0}\\}$.

Hence we can apply Lemma 3.5 to obtain:

  1. (i)

For each $i\in[t]$, every edge $e$ in
$\Gamma\setminus(\bigcup_{j=1}^{i}V_{j}\cup\\{v_{0}\\})$ satisfies $w(e)\in
B_{i}$.

  2. (ii)

We have $k>d_{1}>\ldots>d_{t}$. In particular, $0<d_{i}<k$ for $i\in[t-1]$.

######  Claim 4.2.

There exists $m\in[3,t]$ such that $k-d_{m}\leq 10(k-d_{m-2})$.

###### Proof.

Suppose $k-d_{m}>10(k-d_{m-2})$ for all $m\in[3,t]$. Then, as $1\leq
k-d_{1}\leq k-d_{2}\leq k$,

$k\geq k-d_{t}>10^{t/2-1}(k-d_{2})\geq 10^{t/2-1}\geq k^{5}/10>k,$

a contradiction. ∎

Fix $m$ as in Claim 4.2. Our next goal is to show that there is a small set
$Z$ of vertices from $\Gamma$ such that the weights of all edges not touching
$Z$ lie in a proper subgroup of $\mathbb{Z}_{p}^{k}$. We can then apply our
inductive hypothesis to bound $|V(\Gamma\setminus Z)|$.

For each $i\in[t]$, let
$\tau_{i}:\mathbb{Z}_{p}^{k}\to\mathbb{Z}_{p}^{k}/B_{i}$ be the natural map
sending $a$ to $a+B_{i}$ for $a\in\mathbb{Z}_{p}^{k}$. For a multisubset $X$
of $\mathbb{Z}_{p}^{k}$, let $\tau_{i}(X):=\widetilde{\bigcup}_{x\in
X}\tau_{i}(x)$.

For each $j\in[m-2]$, define $\mathcal{X}_{j}$ to be a minimal subset of
$\mathcal{G}_{j}$ such that
$X_{j}:=\widetilde{\bigcup}_{g\in\mathcal{X}_{j}}g^{*}$ satisfies
$\Sigma(\tau_{m}(X_{j}))=\Sigma(\tau_{m}(U_{j}))$. Similarly, define
$\mathcal{Y}_{j}$ to be a minimal set of gadgets in $\mathcal{G}_{j}$ such
that $Y_{j}:=\widetilde{\bigcup}_{g\in\mathcal{Y}_{j}}g^{*}$ satisfies
$\Sigma(\tau_{m-1}(Y_{j}))=\Sigma(\tau_{m-1}(U_{j}))$. Observe that
$\tau_{m}(X_{j})$ is reduced in $\mathbb{Z}_{p}^{k}/B_{m}$ and
$\tau_{m-1}(Y_{j})$ is reduced in $\mathbb{Z}_{p}^{k}/B_{m-1}$. In particular,

$|\mathcal{X}_{j}|\leq h_{p}(k-d_{m}),\qquad|\mathcal{Y}_{j}|\leq
h_{p}(k-d_{m-1}).$ (4)

Notice that, by definition of $\mathcal{X}_{j}$ and $\mathcal{Y}_{j}$,

$\Sigma(U_{j})\subseteq\Sigma(X_{j}\,\widetilde{\cup}\,U_{m}),\qquad\Sigma(U_{j})\subseteq\Sigma(Y_{j}\,\widetilde{\cup}\,U_{m-1}).$
(5)

Indeed, if $u\in\Sigma(U_{j})$ then there exist $x\in\Sigma(X_{j})$ and $v\in
B_{m}$ such that $u=x+v$, showing
$\Sigma(U_{j})\subseteq\Sigma(X_{j})+B_{m}\subseteq\Sigma(X_{j})+\Sigma(U_{m})=\Sigma(X_{j}\,\widetilde{\cup}\,U_{m})$.
A similar argument holds for $Y_{j}$. Define

$Z:=\\{v_{0}\\}\cup\bigcup_{i\in[m-2]}\big{(}V(\mathcal{X}_{i})\cup
V(\mathcal{Y}_{i})\big{)},$ (6)

where $V(\mathcal{X}_{i})$ denotes the union of vertex sets of gadgets in
$\mathcal{X}_{i}$, and similarly for $V(\mathcal{Y}_{i})$.

######  Claim 4.3.

$|Z|\leq 6m\cdot h_{p}(k-d_{m})$.

###### Proof.

We have

$\displaystyle|Z|$ $\displaystyle\leq
3\sum_{j\in[m-2]}\big{(}|\mathcal{X}_{j}|+|\mathcal{Y}_{j}|\big{)}+1$
$\displaystyle\leq
3\sum_{j\in[m-2]}\big{(}h_{p}(k-d_{m})+h_{p}(k-d_{m-1})\big{)}+1$
$\displaystyle\leq 6m\cdot h_{p}(k-d_{m}),$

where for the first inequality we used that gadgets consist of three vertices,
for the second we used (4), and for the third we used (ii) and the
monotonicity of $h_{p}(\cdot)$. ∎

######  Claim 4.4.

Every edge $e\in E(\Gamma\setminus Z)$ satisfies $w(e)\in B_{m-2}$.

###### Proof.

Suppose there exists an edge $xy$ with no endpoint in $Z$ such that
$w(xy)\notin B_{m-2}$. Define, for convenience,
$V_{\infty}:=V(\Gamma)\setminus\bigcup_{i\in[m]}V_{i}$. Let $g$ be the gadget
rooted at $(v_{0},y)$ with $V(g)=\\{v_{0},x,y\\}$, $P(g)=v_{0}y$, and
$Q(g)=v_{0}xy$. Its value is

$g^{*}=w(Q(g))-w(P(g))=w(v_{0}xy)-w(v_{0}y)=w(v_{0}x)+w(xy)-w(v_{0}y)=w(xy),$

using $w(v_{0}u)=0$ for $u\neq v_{0}$. As $w(xy)\notin B_{m-2}$, it follows
that $g^{*}\notin B_{m-2}$.

We will show that it is possible to find a collection of families of vertex-
disjoint gadgets
$\mathcal{G}^{\prime}_{1},\ldots,\mathcal{G}^{\prime}_{m-3},\mathcal{G}^{\prime\prime}_{m-2}$
in $\Gamma$ with corresponding sets
$U^{\prime}_{1},\ldots,U_{m-3}^{\prime},U^{\prime\prime}_{m-2}$ (where
$U^{\prime}_{i}:=\widetilde{\bigcup}_{g\in\mathcal{G}^{\prime}_{i}}g^{*}$ and
similarly for $U^{\prime\prime}_{m-2}$) satisfying properties (P1) and (P2)
with the additional conditions

  1. (C1)

$|\Sigma(U^{\prime}_{j})|\geq|\Sigma(U_{j})|$ for all $j\in[m-3]$, and

  2. (C2)

$|\Sigma(U^{\prime\prime}_{m-2})|>|\Sigma(U_{m-2})|$.

This will imply that $\mathcal{G}_{1},\ldots,\mathcal{G}_{t}$ does not satisfy
property (P3), as we are able to obtain a sequence that is later in the
lexicographic ordering, a contradiction.

Suppose $x\in V_{a}$ and $y\in V_{b}$, where $a,b\in[m]\cup\\{\infty\\}$ and
$a\leq b$. We need an ad-hoc piece of notation; let
$m^{\prime},m^{\prime\prime}$ satisfy
$\\{m^{\prime},m^{\prime\prime}\\}=\\{m-1,m\\}$ and, if $b\in\\{m-1,m\\}$ but
$a\notin\\{m-1,m\\}$, then $m^{\prime}\neq b$. We define
$\mathcal{G}^{\prime}_{1},\ldots,\mathcal{G}^{\prime}_{m-2}$ as follows.

$\mathcal{G}^{\prime}_{j}:=\begin{cases}\mathcal{G}_{j}&\text{ if
$j\in[m-2]\setminus\\{a,b\\}$,}\\\
\mathcal{X}_{j}\cup\mathcal{Y}_{j}\cup\mathcal{G}_{m^{\prime}}&\text{ if
$j\in[m-2]$ and $j=a$}\\\
\mathcal{X}_{j}\cup\mathcal{Y}_{j}\cup\mathcal{G}_{m^{\prime\prime}}&\text{ if
$j\in[m-2]$ and $j=b>a$.}\end{cases}$

Notice that the gadgets in each $\mathcal{G}_{j}$, with $j\in[m-2]$, are
pairwise vertex-disjoint. Indeed, this holds because
$V(\mathcal{X}_{j}),V(\mathcal{Y}_{j})\subseteq V_{j}$ and the sets $V_{j}$
are pairwise vertex-disjoint. Similarly, any two gadgets from distinct sets
$\mathcal{G}_{j}^{\prime}$ are vertex-disjoint. Additionally, the gadgets in
$\bigcup_{j\in[m-2]}\mathcal{G}_{j}^{\prime}$ are vertex-disjoint from
$(V_{a}\cup V_{b}\cup V_{\infty})\setminus Z$ (if $b\in\\{m-1,m\\}$ and
$a\notin\\{m-1,m\\}$ this follows from the choice of $m^{\prime}$), and hence
they are vertex-disjoint from $V(g)=\\{x,y,v_{0}\\}$. Altogether, writing
$\mathcal{G}_{m-2}^{\prime\prime}:=\mathcal{G}_{m-2}^{\prime}\cup\\{g\\}$,
this shows that
$\mathcal{G}_{1}^{\prime},\ldots,\mathcal{G}_{m-3}^{\prime},\mathcal{G}_{m-2}^{\prime\prime}$
satisfies (P1). Property (P2) holds by construction.

Next, we claim that $\Sigma(U_{j}^{\prime})=\Sigma(U_{j})$ for $j\in[m-2]$,
where
$U^{\prime}_{m-2}:=\widetilde{\bigcup}_{g\in\mathcal{G}^{\prime}_{m-2}}g^{*}$.
This is clear when $j\neq a,b$. Suppose now that
$\mathcal{G}_{j}^{\prime}=\mathcal{X}_{j}\cup\mathcal{Y}_{j}\cup\mathcal{G}_{m-1}$.
If $u\in\Sigma(U_{j})$, then by choice of $\mathcal{Y}_{j}$ there exists
$w\in\Sigma(Y_{j})$ such that $u+B_{m-1}=w+B_{m-1}$. Equivalently, $u-w\in
B_{m-1}$ and thus $u-w\in\Sigma(U_{m-1})$. This implies that
$u\in\Sigma(Y_{j}\,\widetilde{\cup}\,U_{m-1})\subseteq\Sigma(U_{j}^{\prime})$,
as claimed. An analogous argument shows the same when
$\mathcal{G}_{j}^{\prime}=\mathcal{X}_{j}\cup\mathcal{Y}_{j}\cup\mathcal{G}_{m}$.
Hence $|\Sigma(U^{\prime}_{j})|\geq|\Sigma(U_{j})|$ for all $j\in[m-2]$, which
proves (C1).

In particular, $\Sigma(U_{m-2})\subseteq\Sigma(U_{m-2}^{\prime})$. This, the
definition of $B_{m-2}$, and that $g^{*}\notin B_{m-2}$ show that

$\Sigma(U_{m-2}^{\prime\prime})=\Sigma(U_{m-2}^{\prime}\,\widetilde{\cup}\,\\{g^{*}\\})\supseteq\Sigma(U_{m-2}\,\widetilde{\cup}\,\\{g^{*}\\})\supsetneq\Sigma(U_{m-2}).$

Thus (C2) holds. ∎

Claims 4.3 and 4.4 show that, by removing at most $6m\cdot h_{p}(k-d_{m})$
vertices from $\Gamma$, we can obtain a graph whose edge weights are contained
in a subgroup isomorphic to $\mathbb{Z}_{p}^{d_{m-2}}$. But as we presumed
$\Gamma$ has no zero-sum cycle, then $|V(\Gamma\setminus Z)|\leq
f_{p}(d_{m-2})$. Hence

$\displaystyle n=|V(\Gamma)|$ $\displaystyle=|V(Z)|+|V(\Gamma\setminus Z)|$
$\displaystyle\leq 6m\cdot h_{p}(k-d_{m})+f_{p}(d_{m-2})$ $\displaystyle\leq
60\log k\cdot h_{p}\big{(}10(k-d_{m-2})\big{)}+60\log(2d_{m-2})\cdot
h_{p}(10d_{m-2})$ $\displaystyle\leq 60\log(2k)\cdot
h_{p}\big{(}10(k-d_{m-2})\big{)}+60\log(2k)\cdot h_{p}(10d_{m-2})$
$\displaystyle\leq 60\log(2k)\cdot h_{p}(10k).$

where in the third line we used the bounds $m\leq 10\log k$ and $k-d_{m}\leq
10(k-d_{m-2})$ and the monotonicity of $h_{p}(\cdot)$, as well as the
induction hypothesis on $f_{p}(\cdot)$ (using $d_{m-2}\leq k-1$; see (ii)),
and in the last line we applied Observation 2.2(v). ∎

##  5 A few concluding remarks

Our main result shows $f(\mathbb{Z}_{p}^{k})=O(pk(\log k)^{2})$ when $p$ is
prime, with a better bound of $O(k\log k)$ when $p=2$. This is tight up to a
factor which is polylogarithmic in $k$, due to the easy lower bound
$f(\mathbb{Z}_{p}^{k})\geq(p-1)k$. It would be nice to close the gap between
these upper and lower bound.

######  Question 5.1.

Is it true that $f(\mathbb{Z}_{p}^{k})=O(pk)$?

It would also be interesting to determine whether similar bounds hold when $p$
is not prime.

Lemma 4.1 provides a bound on $f_{p}(k)$ in terms of $h_{p}(k)$. In light of
this, one way to improve Theorem 1.1 could be to improve our understanding of
$h_{p}(k)$. Indeed, proving that $h_{p}(k)=O(pk)$ would give an improvement of
a $\log(k)$ factor in Theorem 1.1. Given that we know (see Observation 2.2
(vi)) that $h_{p}(1)=p-1$, a next step towards understanding $h_{p}(k)$ could
be to determine $h_{p}(2)$. Theorem 2.1 gives $h_{p}(2)\leq 4(p-1)$. To this
end, we can prove the following.

######  Lemma 5.2.

Let $p\geq 7$ be prime. Then $h_{p}(2)<\frac{5}{2}(p-1)$.

For completeness, we provide a proof in Appendix A. However, we do not think
that this bound is best possible.

######  Conjecture 5.3.

Let $p$ be prime. Then there exists a constant $C$ such that $h_{p}(2)\leq
2p+C$.

We believe that the proof methods in this paper could be extended to give
bounds on $f(G)$ for other finitely generated abelian groups $G$. It is also a
natural interesting question in its own right to consider the problem of
bounding the size of a largest reduced multiset in such a $G$.

### Acknowledgements

We are very grateful to Lisa Sauermann for coming up with Theorem 2.1 and for
sharing her solution with us; this allowed us to both shorten our paper and
improve our results. We would also like to thank Noga Alon for interesting
discussions regarding reduced sets and for bringing [4] to our attention.
These discussions took place during the workshop ‘Recent advances in
probabilistic and extremal combinatorics’ which was held in Ascona,
Switzerland on 7–12 August 2022. The first author would like to thank Asaf
Shapira and Benny Sudakov, the workshop’s organisers, for inviting her and
giving her the opportunity to give a talk. We would also like to thank the
referees for their helpful comments.

## References

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  * [6] R. Caro, _Zero-sum problems – a survey_ , Discr. Math. 152 (1996), 93–113.
  * [7] S. Das, N. Draganić, and R. Steiner, _Tight bounds for divisible subdivisions_ , J. Combin. Theory Ser. B 165 (2024), 1–19.
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##  Appendix A Proof of Lemma 5.2

The next lemma is a rephrasing of Corollary 3.4 in [4]. We give a similar
proof here.

######  Lemma A.1 (Corollary 3.4 in [4], rephrased).

Let $v_{i}=(v_{i}(1),\ldots,v_{i}(k))$ be a sequence of $k(p-1)$ vectors in
$\mathbb{Z}_{p}^{k}$. Suppose that

$\sum_{(I_{1},\ldots,I_{k})\in\mathcal{I}}\,\,\prod_{i\in[k]}\prod_{j\in
I_{i}}v_{j}(i)\neq 0,$ (7)

where $\mathcal{I}$ is the collection of equipartitions of $[k(p-1)]$. Then
$\Sigma(\\{v_{1},\ldots,v_{k(p-1)}\\})=\mathbb{Z}_{p}^{k}$.

Our proof uses Alon’s Combinatorial Nullstellensatz.

######  Theorem A.2 (Alon’s Combinatorial Nullstellensatz [2]).

Let $f\in\mathbb{F}[x_{1},\ldots,x_{n}]$ be a polynomial with coefficients in
a field $\mathbb{F}$ such that the degree of $f$ is $\sum_{i=1}^{n}t_{i}$ and
the coefficient of $\prod_{i=1}^{n}x_{i}^{t_{i}}$ is non-zero. Let
$S_{1},\ldots,S_{n}$ be subsets of $\mathbb{F}$ such that $|S_{i}|>t_{i}$ for
all $i$. Then there exists $(s_{1},\ldots,s_{n})\in S_{1}\times\ldots\times
S_{n}$ such that $f(s_{1},\ldots,s_{n})\not=0$.

###### Proof of Lemma A.1.

Fix $w=(w(1),\ldots,w(k))\in\mathbb{Z}_{p}^{k}$. Set $m:=k(p-1)$ and define

$P(x_{1},\ldots,x_{m}):=\prod_{i=1}^{k}\left(\bigg{(}\sum_{j\in[m]}x_{j}v_{j}(i)-w(i)\bigg{)}^{p-1}-1\right).$
(8)

Let $S$ be the multiset $\widetilde{\bigcup}_{i\in[m]}v_{i}$. Observe that
$P(y)\neq 0$ for some $y\in\\{0,1\\}^{m}$ if and only if $w\in\Sigma(S)$ (via
Fermat’s little theorem).

Thus, applying Theorem A.2, with $S_{i}=\\{0,1\\}$ and $t_{i}=1$ for every
$i\in[m]$, we see that if the coefficient of $\prod_{i\in[m]}x_{i}$ in $P$ is
non-zero then $w\in\Sigma(S)$. This coefficient is the coefficient of
$\prod_{i\in[m]}x_{i}$ in
$\prod_{i=1}^{k}\left(\sum_{j\in[m]}x_{j}v_{j}(i)\right)^{p-1}$. In order to
obtain a term of $\prod_{i\in[m]}x_{i}$ from the latter product, from each
factor we must select a distinct variable $x_{j}$ in such a way that every
$j\in[m]$ appears exactly once. Recalling that $\mathcal{I}$ is the collection
of equipartitions $(I_{1},\ldots,I_{k})$ of $[m]$ and thinking of $I_{i}$ as
indexing the variables obtained from the $i$th factor of the product, this
coefficient is

$\big{(}(p-1)!\big{)}^{k}\sum_{(I_{1},\ldots,I_{k})\in\mathcal{I}}\,\,\prod_{i\in[k]}\,\prod_{j\in
I_{i}}v_{j}(i).$

As $p$ is prime, $(p-1)!\neq 0$, and (7) holds if and only if the coefficient
of $\prod_{i\in[m]}x_{i}$ in $P$ is non-zero, which implies that
$w\in\Sigma(S)$. Since $w$ was an arbitrary element in $\mathbb{Z}_{p}^{k}$,
this shows that (8) implies $\Sigma(S)=\mathbb{Z}_{p}^{k}$. ∎

We now prove Lemma 5.2, restated here. See 5.2

###### Proof of Lemma 5.2.

Suppose that $T\>\widetilde{\subseteq}\>\mathbb{Z}_{p}^{2}$ is reduced and
$|T|\geq\frac{5}{2}(p-1)$. Let $S\subseteq T$ satisfy $|S|=\frac{5}{2}(p-1)$.
By Observation 2.2 (iii), $S$ is reduced. By Observation 2.2 (i), $(0,0)\notin
S$. We will show that $\Sigma(S)=\mathbb{Z}_{p}^{2}$, which is a contradiction
as this implies $T$ is not reduced.

For $v\in\mathbb{Z}_{p}^{2}\setminus\\{(0,0)\\}$, define $\langle
v\rangle:=\\{\alpha v:\alpha\in\mathbb{Z}_{p}\\}$. Choose $v$ to minimise the
size of the multiset intersection $\langle v\rangle\,\widetilde{\cap}\,S$,
defined to be the multisubset of $S$ whose elements lie in $\langle v\rangle$.
Notice that there are $p+1$ different ‘directions’, namely $(1,i)$ for
$i\in[0,p-1]$ and $(0,1)$, and each $v\in\mathbb{Z}_{p}^{k}$ lies in $\langle
d\rangle$ for exactly one direction $d$. Hence, as
$|S|=\frac{5}{2}(p-1)<3(p+1)$, we have $|\langle
v\rangle\,\widetilde{\cap}\,S|\leq 2$. Let
$f:\mathbb{Z}_{p}^{2}\to\mathbb{Z}_{p}^{2}$ be an injective linear
transformation mapping $v$ to $(0,1)$. Then, by Observation 2.2 (iv), $f(S)$
is a reduced multisubset of $\mathbb{Z}_{p}^{2}$ of size $\frac{5}{2}(p-1)$
with the property that $f(S)$ contains at most two vectors in direction
$(0,1)$. That is, the multiset intersection
$f(S)\,\widetilde{\cap}\,\langle(0,1)\rangle$ has size at most 2.

Let $S_{0}$ be obtained from $S$ by removing all elements in
$\langle(0,1)\rangle$, and observe that $|S_{0}|\geq\frac{5}{2}(p-1)-2>p-1$.
Let $v_{1},\ldots,v_{p-1}$ be chosen as follows, defining
$S_{i}:=S_{0}-\\{v_{1},\ldots,v_{i}\\}$ once $v_{1},\ldots,v_{i}$ are defined.
Having chosen $v_{1},\ldots,v_{i-1}$, choose $v_{i}\in S_{i-1}$ to maximise
$|\langle v_{i}\rangle\,\widetilde{\cap}\,S_{i-1}|$.

Write $S^{\prime}:=S-\\{v_{1},\ldots,v_{p-1}\\}$ and define

$m:=\max_{v\in\mathbb{Z}_{p}^{2}\setminus\\{(0,0)\\}}|S^{\prime}\,\widetilde{\cap}\,\langle
v\rangle|.$ (9)

######  Claim A.3.

$m\leq(p-1)/2$.

###### Proof.

Let $T$ be the set of elements $t\in\mathbb{Z}_{p}$ such that
$\langle(1,t)\rangle\cap\\{v_{1},\ldots,v_{p-1}\\}\neq\emptyset$. Then, by
definition of $v_{1},\ldots,v_{p-1}$, we have
$|S^{\prime}\,\widetilde{\cap}\,\langle(1,t)\rangle|\in\\{m-1,m\\}$ for every
$t\in T$.

If $|T|\leq 2$ then

$2(p-1)\geq|S\,\widetilde{\cap}\,\bigcup_{t\in T}\langle(1,t)\rangle|\geq
p-1+m+m-1,$

using that $|S\,\widetilde{\cap}\,\langle v\rangle|\leq p-1$ for
$v\in\mathbb{Z}_{p}^{2}\setminus\\{(0,0)\\}$ (which follows from Observation
2.2 (vi)). This implies that $m\leq p/2$, and thus $m\leq(p-1)/2$ because $p$
is odd.

If $|T|\geq 3$ then

$\frac{5}{2}(p-1)\geq\left|S\cap\widetilde{\bigcup}_{t\in
T}\langle(1,t)\rangle\right|\geq p-1+m+(|T|-1)(m-1)\geq p+3m-3.$

Thus $m\leq p/2+1/6$, implying that $m\leq(p-1)/2$, as $p$ is odd. This
completes the proof of Claim A.3. ∎

######  Claim A.4.

For every $t\in[0,p-1]$ there exists a multisubset
$\widetilde{\bigcup}_{i\in[p-1+t]}v_{i}\subseteq S^{\prime}$ such that

$\sum_{(I_{1},I_{2})\in\mathcal{I}_{t}}\,\,\prod_{i\in[2]}\prod_{j\in
I_{i}}v_{j}(i)\neq 0,$ (10)

where $\mathcal{I}_{t}$ is the collection of partitions $(I_{1},I_{2})$ of
$[p-1+t]$ such that $|I_{1}|=p-1$.

###### Proof.

We prove the claim by induction on $t$. When $t=0$, the expression in (10) is
$\prod_{j\in I_{1}}v_{j}(1)$, which is non-zero as
$v_{1},\ldots,v_{p-1}\notin\langle(0,1)\rangle$.

Now suppose that $t\in[p-1]$ and $v_{1},\ldots,v_{p-1+t-1}$ satisfy the
requirements of the claim for $t-1$. Expanding the left-hand side of (10),
with $v_{p-1+t}=(x,y)$, gives

$xs_{1}+ys_{2},$

where $s_{1}$ is a sum of terms depending on $v_{1},\ldots,v_{p-1+t-1}$ and

$s_{2}=\sum_{(I_{1},I_{2})\in\mathcal{I}_{t-1}}\,\,\prod_{i\in[2]}\prod_{j\in
I_{i}}v_{j}(i)\neq 0.$

The multisubset of $S-\\{v_{1},\ldots,v_{p-1+t-1}\\}$ of vectors $(x,y)$
satisfying $xs_{1}+ys_{2}=0$ is contained in a subspace of dimension $1$.
Because $m\leq(p-1)/2$ (by Claim A.3) and
$|S-\\{v_{1},\ldots,v_{p-1+t-1}\\}|\geq(p+1)/2$, we find that there is a
suitable $v_{p-1+t}\in S-\\{v_{1},\ldots,v_{p-1+t-1}\\}$. ∎

Lemma A.1 and Claim A.4 (with $t=p-1$) imply that there is a multisubset
$S^{\prime}\subseteq S$ of size $2(p-1)$ such that
$\Sigma(S^{\prime})=\mathbb{Z}_{p}^{2}$. As $S^{\prime}\not=S$, this
contradicts the assumption that $S$ is reduced. ∎


# Anisotropy of upper critical fields and interface superconductivity in
FeSe/SrTiO3 grown by PLD

Tomoki Kobayashi, Hiroki Nakagawa, Hiroki Ogawa, Fuyuki Nabeshima, and
Atsutaka Maeda  Dept. of Basic Science, University of Tokyo, Meguro, 153-8902,
Tokyo, Japan<EMAIL_ADDRESS>
###### Abstract

In this study, we grow FeSe/SrTiO3 with thicknesses of 4–19 nm using pulsed
laser deposition and investigate their magneto-transport properties. The
thinnest film (4 nm) exhibit negative Hall effect, indicating electron
transfer into FeSe from the SrTiO3 substrate. This is in agreement with
reports on ultrathin FeSe/SrTiO3 grown by molecular beam epitaxy. The upper
critical field is found to exhibit large anisotropy ($\gamma>$ 11.9),
estimated from the data near the transition temperature ($T_{\mathrm{c}}$). In
particular, the estimated coherence lengths in the perpendicular direction are
0.15–0.27 nm, which are smaller than the c-axis length of FeSe, and are found
to be almost independent of the total thicknesses of the films. These results
indicate that superconductivity is confined at the interface of FeSe/SrTiO3.

##  1 Introduction

Monolayer FeSe on SrTiO3 (STO) shows a significantly enhanced superconducting
transition temperature ($T_{\mathrm{c}}$) [1, 2] from that of 9 K in the bulk
form [3], which was claimed by the opening of the superconducting gap-like
structure at a temperature as high as 65 K observed by the angle-resolved
photoemission spectroscopy (ARPES) measurements [1, 2]. The $T_{\mathrm{c}}$
enhancement is mainly attributed to the interfacial electron-phonon coupling
with optical phonons of STO [4] and/or electron transfer from the STO
substrate [5]. However, zero-resistivity has been reported only below 30 K by
transport measurements [1, 6, 7], except for the singular report of
$T_{\mathrm{c}}$ $>$ 100 K [8]. Although the large difference between the
spectroscopic $T_{\mathrm{c}}$ and the transport $T_{\mathrm{c}}$ has been
discussed in terms of the superconducting fluctuation in two-dimensional
systems[7], zero resistivity below 46 K is realized even in surface-electron-
doped FeSe films by the electric-double-layer transistor (EDLT) technique [9,
10, 11], suggesting the presence of further space for the improvement of
transport $T_{\mathrm{c}}$ of the interface superconductivity in FeSe/STO. For
that purpose, systematic investigation of the interface structure for a wide
range of combinations with oxide materials must be necessary, and we believe
that pulsed laser deposition (PLD) is a suitable growth technique. Indeed, we
fabricated FeSe films on the STO substrate whose surface was atomically flat
and found that the interface superconductivity was realized even in the PLD-
grown films, based on the strain vs. $T_{\mathrm{c}}$ relation and the film
thickness vs. $T_{\mathrm{c}}$ relation [12]. However, stronger evidence of
the interface superconductivity in the PLD-grown films is desired.

In this study, we grow FeSe/STO with thicknesses of 4–19 nm using PLD and
investigated their magneto-transport properties. The thinnest film (4 nm)
exhibits negative Hall effect, indicating electron transfer into FeSe from the
STO substrate. This is in agreement with reports on ultrathin FeSe/STO grown
by molecular beam epitaxy (MBE) [13, 14]. The upper critical field estimated
from the data near $T_{\mathrm{c}}$ was found to exhibit large anisotropy
($\gamma\sim$ 11 – 25). In particular, the estimated coherence lengths in the
perpendicular direction are 0.15–0.27 nm, which is smaller than the c-axis
length of FeSe, and are almost independent of the total thickness of the film.
These results indicate that superconductivity is confined to the interface of
FeSe/STO.

##  2 Experimental methods

FeSe films were grown on atomically flat STO (100) substrates using the PLD
technique[12]. The preparation of an atomically flat surface of the STO
substrate is essential for realizing the interface superconductivity in this
study. Ultrathin films of FeSe deteriorate easily in air, leading to the
disappearance of superconductivity. In order to protect the films from
degradation caused by the air-exposure, we deposited amorphous Si on FeSe/STO
at room temperature also by PLD. In order to obtain electrical contact with
the Si-deposited FeSe/STO, FeSe was deposited as electrodes at room
temperature prior to the growth of the crystalline FeSe film. The orientation
and the crystal structure of the grown films were characterized using X-ray
diffraction (XRD) measurements with Cu K$\alpha$ radiation at room
temperature. The thicknesses of thick films were estimated by the Laue fringes
of the (001) reflection of FeSe. As for the ultrathin films, where FeSe
reflections were not observed due to weak intensities, the film thicknesses
were estimated by the growth rate, which is always monitored for all films
including thicker ones. Resistivity and Hall measurements were conducted using
a physical property measurement system (PPMS) from 2 to 300 K.

##  3 Results and discussion

Figures 1(a) and 1(b) show the temperature dependence of the sheet resistance
of samples #1 and #2 with thicknesses of 4 and 19 nm, respectively. Both
samples showed superconductivity at low temperatures. In the previous study
using PLD [12], even a 5-nm-thick film did not show superconductivity possibly
because of the degradation by the air exposure. The appearance of
superconductivity in sample #1 (4 nm) indicates that the capping layer of Si
successfully protects the sample from the air exposure. The onset
$T_{\mathrm{c}}$ ($T_{\mathrm{c}}^{\mathrm{onset}}$ ) of the thinner sample #1
(4 nm) was 30 K and higher than that of sample #2 (19 nm)
($T_{\mathrm{c}}^{\mathrm{onset}}$ = 26 K). This is in agreement with our
previous results[12] and is a common trend among ultrathin FeSe/STO including
those grown by MBE[13, 14], which is considered to be one of the hallmarks of
the interface superconductivity.  One might think that superconductivity
properties are the same between the thinner film and the thicker film if
superconductivity takes place only at the interface, which is different from
our observation. We consider the origin of the difference to be due to the
difference in the carrier density near the interface. We will discuss this
issue later again. The transition of sample #1 (4 nm) was broader than that of
sample #2 (19 nm), which indicates more inhomogeneity in the thinner film. We
consider the reason for the observed more inhomogeneous behavior in the
thinner film as follows. We deposited amorphous Si on the FeSe surface for the
protection of FeSe by the PLD technique. Since amorphous Si deposited by the
PLD has high kinetic energy[15], this can damage an area down to a few nm from
the FeSe surface. Thus thinner films can be more strongly damaged, leading to
the observed behavior.

Figure 1: (a) Temperature dependence of the sheet resistance of sample #1 (4
nm) and sample #2 (19 nm), and (b) the expanded plot in the vicinity of
$T_{c}$. The sheet resistance is normalized by that at 45 K. (c) Magnetic-
field dependence of Hall resistance $R_{xy}$ of sample #1 (4 nm) at different
temperatures and (d) those for sample #2 (19 nm). (e) temperature dependence
of $R_{xy}/B$ of sample #1 (4 nm) and sample #2 (19 nm). $R_{xy}/B$ is equal
to $R_{H}/d$, where $R_{H}$ and $d$ are the Hall coefficient and the film
thickness, respectively.

Figures 1(c)–(e) show the temperature dependence of the Hall resistance
$R_{xy}$ as a function of magnetic field $B$. As shown in Fig. 1(c) and 1(d),
both films showed linear dependence of Hall resistance $R_{\mathrm{xy}}$ on
magnetic fields from 300 K to 40 K. In particular, in sample #1 (4 nm),
$R_{\mathrm{xy}}$ showed a negative gradient with $B$ below 200 K, while it
became positive at room temperature (Fig. 1(c)), which can be recognized more
easily in Fig.1(e), where $R_{\mathrm{xy}}/B$ (which is equal to
$R_{\mathrm{H}}/d$, where $R_{\mathrm{H}}$ and $d$ is the Hall coefficient and
the film thickness, respectively) are plotted as a function of temperature.
This is in contrast to the Hall data in FeSe flakes[16] and thin films grown
on substrates without any special treatment[17], where $R_{\mathrm{H}}$ shows
a positive sign even below 150 K. Our result for sample #1 (4 nm) is rather
similar to the MBE-grown FeSe/STO where $R_{\mathrm{H}}$ exhibits negative
signs at low temperatures, exhibiting the electron transfer into FeSe from the
STO substrate[13, 14]. Thus, the above results showed that the carrier
transfer from the STO substrate takes place even in the PLD-grown FeSe/STO
films as well as in the MBE-grown films. In contrast, sample #2 (19 nm) shows
the positive $R_{\mathrm{xy}}/B$ even at low temperatures, and the temperature
dependence of $R_{\mathrm{xy}}/B$ is similar to those of FeSe films without
the substrate treatment[17], although it shows ”high” $T_{\mathrm{c}}$ which
is characteristic of the interface superconductivity. The difference in
$R_{\mathrm{xy}}/B$ between sample #1 (4 nm) and sample #2 (19 nm) can be
understood as follows.

Electron transfer is limited to a few layers near the FeSe/STO interface in
sample #1 (4 nm), whereas in the thicker sample sample #2 (19 nm) transferred
carrier is diluted, leading to the positive $R_{\mathrm{xy}}/B$ as a whole.
Thus, thicker films showed positive $R_{\mathrm{xy}}/B$ even at low
temperatures. This also explains why superconductivity properties are
different between the two samples even when superconductivity takes place at
the interface. The difference in the carrier density can cause the difference
in superconductivity properties. Indeed, a similar discussion is found in a
paper[14], where they investigated a series of FeSe/STO films with different
thicknesses. In addition to the transport data such as resistivity and Hall
effects, the authors used atomically resolved electron energy-loss
spectroscopy (EELS) and concluded that the superconductivity is confined at
the interface regardless of FeSe thickness, although $T_{\mathrm{c}}$ differs
due to the difference in the number of electrons at the interface.

Figure 2: (a),(b) Temperature dependence of sheet resistance of sample #1 (4
nm) for $B$$/\\!/\textit{c}$ and $B$$/\\!/\textit{ab}$, respectively. (c)
Temperature dependence of the upper critical fields of sample #1 (4 nm) for
$B$$/\\!/\textit{c}$, $B_{c2}^{\perp}$ (red) and $B$$/\\!/\textit{ab}$,
$B_{c2}^{/\\!/}$ (blue). (d),(e) Temperature dependence of sheet resistance of
sample #2 (19 nm) for $B$$/\\!/\textit{c}$ and $B$$/\\!/\textit{ab}$,
respectively. (f) Temperature dependence of $B_{c2}^{\perp}$ (red) and
$B_{c2}^{/\\!/}$ (blue) of sample #2 (19 nm).

To investigate the anisotropy of superconductivity in the grown films, we
conducted magneto-transport measurements on the grown films across
$T_{\mathrm{c}}$. Figure 2 shows the temperature dependence of the sheet
resistance of the films under magnetic fields up to 9 T for
$B$$/\\!/\textit{c}$ and $B$$/\\!/\textit{ab}$. For $B$$/\\!/\textit{c}$,
$T_{\mathrm{c}}^{\mathrm{onset}}$ of sample #1 (4 nm) decreases slightly from
26 K to 20 K, and the superconducting transition became broader with
increasing field (Fig. 2(a)). One the other hand, for $B$$/\\!/\textit{ab}$
(Fig. 2(b)), $T_{\mathrm{c}}^{\mathrm{onset}}$ was not changed almost at all,
and a little broadening was observed near $T_{\mathrm{c}}^{\mathrm{zero}}$,
suggesting a very high upper critical field $B_{\mathrm{c2}}$.
$B_{\mathrm{c2}}(T)$ for $B$$/\\!/\textit{c}$ ($B_{c2}^{\perp}(T)$) and
$B$$/\\!/\textit{ab}$ ($B_{c2}^{/\\!/}(T)$) was determined by taking a
criterion of the field at 50% of the normal-state resistance
$R_{\mathrm{n}}(T)$ (Fig.2(c)). With the aid of the
Werthamer–Helfand–Hohenberg (WHH) formula i.e. $B_{c2}(0)=-0.69T_{\mathrm{c}}\
|dB_{\mathrm{c2}}/dT_{\mathrm{c}}|$, where the only orbital pair breaking
effect is considered, and setting $T_{\mathrm{c}}$ =
$T_{\mathrm{c}}^{\mathrm{onset}}$ , we obtained $B_{c2}(0)$. Note that the
paramagnetic effect and the spin-orbital coupling should be considered to
estimate actual $B_{c2}(0)$. Indeed, even in an early publication on bulk
single crystals of FeSe, the upper critical field becomes almost isotropic at
low temperatures, which is considered to be due to the paramagnetic pair
braking[18]. However, the above analysis is more suitable to discuss
anisotropiy of the coherence length of superconductivity origin. We obtained
$B_{c2}^{\perp}(0)=45.8$ T and $B_{c2}^{/\\!/}(0)$=675 T for sample #1 (4 nm),
providing a very large anisotropy parameter
$\gamma=B_{c2}^{/\\!/}(0)/B_{c2}^{\perp}(0)=14.8$. Similar data for sample #2
(19 nm) shown in Fig. 2(d–f) and the same analysis also exhibited a large
anisotropy with $\gamma=11.9$. These values of $\gamma$ are larger than those
of much thicker FeSe films grown on the substrate without any special
treatment ($\gamma\sim 2.5)$)[19] and those of intercalated FeSe systems with
an expanded distance between superconducting FeSe layers ($\gamma\leq
11)$)[20, 21]. The GL coherence lengths of sample #1 (4 nm) are
$\xi_{ab}(0)=\sqrt{\phi_{0}/2\pi B_{c2}^{\perp}}$ = 2.68 Å and
$\xi_{c}(0)=\sqrt{\phi_{0}/2\pi B_{c2}^{/\\!/}\xi_{ab}}$ = 0.18 Å. The
obtained $\xi_{ab}(0)$ is similar to that of monolayer FeSe/STO grown by
MBE[22]. Interestingly, the estimated $\xi_{c}(0)$ is shorter than the
distance between FeSe layers, suggesting superconductivity is confined in very
thin layers.

To look for the origin of the extremely confined superconductivity, we
investigated $\xi_{c}(0)$ for various films with different thicknesses (Fig.
3). $\xi_{c}(0)$ were 0.15–0.27 nm, which are also shorter than the distance
between FeSe layers, and were found to show a very weak dependence on the film
thickness. This result suggests that the presence of the FeSe/STO interface is
essential and that the superconductivity is confined at the FeSe/STO
interface.

Figure 3: The GL coherence lengths $\xi_{\mathrm{c}}(0)$ as a function of the
film thckness.

It is suggestive to compare our result with those of thin flakes of FeSe[16],
where $\xi_{c}$ was investigated for the thickness in the range 9–100 nm. For
thick flakes, $xi_{c}$ was $\sim$ 1–2 nm, whereas it decreases down to
$\sim$0.6 nm with decreasing the thickness of the flake, which the authors of
ref.[16] claimed was evidence that superconductivity is confined to one FeSe
layer in the thin-flake limit. On the other hand, in our thin films,
$\xi_{c}\sim$0.15–0.27 nm, which is smaller than FeSe-FeSe distance and is
nearly independent of the total film thickness. This suggests that the
superconductivity observed in our ultrathin films is qualitatively different
from those observed in the thin flakes, and again suggests that it is confined
at the interface between FeSe and the STO substrate.

Therefore, together with a negative Hall effect in thinner films, the above-
mentioned very short $\xi_{\mathrm{c}}(0)$ which was almost independent of the
total film thickness is additional evidence that the interface
superconductivity is realized in our PLD-grown films.

##  4 Conclusion

In conclusion, we grew FeSe/STO with different thicknesses of 4–19 nm using
PLD and investigated the magneto-transport effect, namely the Hall effect and
the the upper critical fields for $B$$/\\!/\textit{c}$ and $B$//ab. The
thinnest film (4 nm thickness) showed negative $R_{xy}/d$ even down to low
temperatures, indicating the electron transfer from the STO substrate. The
anisotropy of the upper critical fields estimated from the data near
$T_{\mathrm{c}}$ was very large ($\gamma\geq$ 11.9), and the estimated c-axis
coherence length, $\xi_{c}(0)$ is very short ($\leq$ 0.27 Å) and almost
independent of the total film thickness. All of these results indicate that
superconductivity is confined at the interface of FeSe/STO in our PLD-grown
films. Thus, the next step is to realize higher zero-resistance
$T_{\mathrm{c}}$ of this interface superconductivity, which is under
investigation.

We would like to acknowledge K. Ueno at the University of Tokyo for their
technical support of the XRD measurements.

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