#  $\mathcal{L}$-intersecting or Configuration Forbidden Families
on Set Systems and Vector Spaces over Finite Fields

Jiuqiang Liua,b,, Guihai Yua,∗, Lihua Fengc, and Yongtao Lic
a College of Big Data Statistics, Guizhou University of Finance and Economics
Guiyang, Guizhou, 550025, China
b Department of Mathematics, Eastern Michigan University
Ypsilanti, MI 48197, USA
c School of Mathematics and Statistics, HNP-LAMA, Central South University
Changsha, Hunan, 410083, China
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>The
corresponding author

###### Abstract

In this paper, we derive a tight upper bound for the size of an intersecting
$k$-Sperner family of subspaces of the $n$-dimensional vector space
$\mathbb{F}_{q}^{n}$ over finite field $\mathbb{F}_{q}$ which gives a
$q$-analogue of the Erdős’ $k$-Sperner Theorem, and we then establish a
general relationship between upper bounds for the sizes of families of subsets
of $[n]=\\{1,2,\dots,n\\}$ with property $P$ and upper bounds for the sizes of
families of subspaces of $\mathbb{F}_{q}^{n}$ with property $P$, where $P$ is
either $\mathcal{L}$-intersecting or forbidding certain configuration.
Applying this relationship, we derive generalizations of the well known
results about the famous Erdős matching conjecture and Erdős-Chvátal simplex
conjecture to linear lattices.

Key words: Erdős-Ko-Rado theorem, Erdős matching conjecture; Erdős-Chvátal
simplex conjecture; hereditary property; $\mathcal{L}$-intersecting family;
Sperner family
AMS Classifications: 05D05.

##  1 Introduction

Throughout the paper, we denote $[n]=\\{1,2,\dots,n\\}$. A family
$\mathcal{F}$ of subsets of $[n]$ is called $t$-$intersecting$ if $|E\cap
F|\geq t$ for every pair of distinct subsets $E,F\in\mathcal{F}$, and it is
called intersecting when $t=1$. Let
$\mathcal{L}=\\{l_{1},l_{2},\dots,l_{s}\\}$ be a set of $s$ nonnegative
integers with $0\leq l_{1}<\cdots<l_{s}$. A family $\mathcal{F}$ of subsets of
$[n]$ is called $\mathcal{L}$-$intersecting$ if $|E\cap F|\in\mathcal{L}$ for
every pair of subsets $E$, $F$ in $\mathcal{F}$. $\mathcal{F}$ is $k$-uniform
if it is a collection of $k$-subsets of $[n]$. Thus, a $k$-uniform
intersecting family is $\mathcal{L}$-intersecting for
$\mathcal{L}=\\{1,2,\dots,k-1\\}$. A $k$-uniform $t$-intersecting family is a
special $\mathcal{L}$-intersecting family with
$\mathcal{L}=\\{t,t+1,\dots,k-1\\}$. A Boolean lattice $\mathcal{B}_{n}$ is
the set $2^{[n]}$ of all subsets of $[n]$ with the ordering being the
containment relation and a linear lattice $\mathcal{L}_{n}(q)$ is the set of
all subspaces of the $n$-dimensional vector space $\mathbb{F}_{q}^{n}$ over
the finite field $\mathbb{F}_{q}$ with the ordering being the relation of
inclusion of subspaces.

In 1961, Erdős, Ko, and Rado [11] proved the following classical result.

###### Theorem 1.1

(Erdős, Ko, and Rado, [11]). Let $n\geq 2k$, and let $\mathcal{A}$ be a
$k$-uniform intersecting family of subsets of $[n]$. Then
$|\mathcal{A}|\leq{{n-1}\choose{k-1}}$, with equality only when $\mathcal{F}$
consists of all $k$-subsets containing a common element.

Since then, many intersection theorems have appeared in the literature, see,
for example, the next three well-known theorems.

###### Theorem 1.2

(Ray-Chaudhuri and Wilson, [28]). Let
$\mathcal{L}=\\{l_{1},l_{2},\dots,l_{s}\\}$ be a set of $s$ nonnegative
integers. If $\mathcal{A}$ is a $k$-uniform $\mathcal{L}$-intersecting family
of subsets of $[n]$, then $|\mathcal{A}|\leq{{n}\choose{s}}$.

###### Theorem 1.3

(Frankl and Wilson, [14]). Let $\mathcal{L}=\\{l_{1},l_{2},\dots,l_{s}\\}$ be
a set of $s$ nonnegative integers. If $\mathcal{A}$ is an
$\mathcal{L}$-intersecting family of subsets of $[n]$, then

$|\mathcal{A}|\leq{{n}\choose{s}}+{{n}\choose{s-1}}+\cdots+{{n}\choose{0}}.$

In terms of the parameters $n$ and $s$, the inequality in Theorem 1.3 is best
possible, as shown by the set of all subsets of $[n]$ with sizes at most $s$
and taking $\mathcal{L}=\\{0,1,\dots,s-1\\}$.

###### Theorem 1.4

(Alon, Babai, and Suzuki, [2]). Let
$\mathcal{L}=\\{l_{1},l_{2},\dots,l_{s}\\}$ be a set of $s$ nonnegative
integers and $K=\\{k_{1},k_{2},\dots,k_{r}\\}$ be a set of integers satisfying
$k_{i}>s-r$ for every $i$. Suppose that
$\mathcal{A}=\\{A_{1},A_{2},\dots,A_{m}\\}$ is a family of subsets of $[n]$
such that $|A_{i}|\in K$ for every $1\leq i\leq m$ and $|A_{i}\cap
A_{j}|\in\mathcal{L}$ for every pair $i\neq j$. Then

$m\leq{{n}\choose{s}}+{{n}\choose{s-1}}+\cdots+{{n}\choose{s-r+1}}.$

Recall that a family of subsets of $[n]$ is called a Sperner family (or
antichain) if there are no two different members of the family such that one
of them contains the other, and a family $\mathcal{F}$ of subsets of $[n]$ is
$k$-$Sperner$ if all chains in $\mathcal{F}$ have length at most $k$. Define
$\sum(n,k)$ to be the sum of the $k$ largest binomial coefficients of order
$n$, i.e.,
$\sum(n,k)=\sum_{i=1}^{k}{{n}\choose{\lfloor\frac{n-k}{2}\rfloor+i}}$. Let
$\sum^{*}(n,k)$ be the collection of families consisting of the corresponding
full levels, i.e., if $n+k$ is odd, then $\sum^{*}(n,k)$ contains one family
$\cup_{i=1}^{k}{{[n]}\choose{\lfloor\frac{n-k}{2}\rfloor+i}}$ (where
${{[n]}\choose{k}}$ is the set of all $k$-subsets of $[n]$); if $n+k$ is even,
then $\sum^{*}(n,k)$ contains two families of the same size
$\cup_{i=0}^{k-1}{{[n]}\choose{\frac{n-k}{2}+i}}$ and
$\cup_{i=1}^{k}{{[n]}\choose{\frac{n-k}{2}+i}}$. The following theorem by
Erdős [9] generalizes the classical Sperner theorem.

###### Theorem 1.5

(Erdős, [9]). Suppose that $\mathcal{A}$ is a $k$-Sperner family of subsets of
$[n]$. Then

$|\mathcal{A}|\leq\sum(n,k).$

Moreover, if $|\mathcal{A}|=\sum(n,k)$, then $\mathcal{A}\in\sum^{*}(n,k)$.

Throughout this paper, we denote the Gaussian (or $q$-binomial) coefficient of
order $n$ by

$\genfrac{[}{]}{0.0pt}{}{n}{k}=\prod_{0\leq i\leq
k-1}\frac{q^{n-i}-1}{q^{k-i}-1}.$ (1)

It is well known that the number of all $k$-dimensional subspaces of
$\mathbb{F}_{q}^{n}$ is equal to $\left[n\atop k\right]$.

Results on the sizes of families of subspaces of $\mathbb{F}_{q}^{n}$
analogous to those theorems on $\mathcal{L}$-intersecting families of subsets
of $[n]$ have appeared in the literature. The next generalization of Erdős-Ko-
Rado Theorem (Theorem 1.1) is proved by Greene and Kleitman [16] and Hsieh
[18].

###### Theorem 1.6

(Greene and Kleitman [16], Hsieh [18]). Let $n\geq 2k+1$. Suppose that
$\mathcal{V}$ is a family of $k$-dimensional subspaces of $\mathbb{F}_{q}^{n}$
satisfying that $\dim(V_{i}\cap V_{j})>0$ for any distinct subspaces $V_{i}$
and $V_{j}$ in $\mathcal{V}$. Then

$|\mathcal{V}|\leq\genfrac{[}{]}{0.0pt}{}{n-1}{k-1}$

and equality holds if and only if there exists a $1$-dimensional subspace $R$
of $\mathbb{F}_{q}^{n}$ such that $R\subseteq V$ for every $V\in\mathcal{V}$.

Frankl and Graham [13] derived the following theorem on
$\mathcal{L}$-intersecting family of $k$-dimensional subspaces of
$\mathbb{F}_{q}^{n}$.

###### Theorem 1.7

(Frankl and Graham [13]). Let $\mathcal{L}=\\{l_{1},l_{2},\dots,l_{s}\\}$ be a
set of $s$ nonnegative integers. Suppose that $\mathcal{V}$ is a family of
$k$-dimensional subspaces of $\mathbb{F}_{q}^{n}$ such that $\dim(V_{i}\cap
V_{j})\in\mathcal{L}$ for any distinct subspaces $V_{i}$ and $V_{j}$ in
$\mathcal{V}$. Then

$|\mathcal{V}|\leq\genfrac{[}{]}{0.0pt}{}{n}{s}.$

In 1990, Lefmann [23] proved the following $\mathcal{L}$-intersecting theorem
for ranked finite lattices, and in 2001, Qian and Ray-Chaudhuri [27] extended
to quasi-polynomial semi-lattices.

###### Theorem 1.8

(Lefmann [23], Qian and Ray-Chaudhuri [27]). Let
$\mathcal{L}=\\{l_{1},l_{2},\dots,l_{s}\\}$ be a set of $s$ nonnegative
integers. Suppose that $\mathcal{V}$ is a family of subspaces of
$\mathbb{F}_{q}^{n}$ satisfying that $\dim(V_{i}\cap V_{j})\in\mathcal{L}$ for
any distinct subspaces $V_{i}$ and $V_{j}$ in $\mathcal{V}$. Then

$|\mathcal{V}|\leq\genfrac{[}{]}{0.0pt}{}{n}{s}+\genfrac{[}{]}{0.0pt}{}{n}{s-1}+\cdots+\genfrac{[}{]}{0.0pt}{}{n}{0}.$

Alon et al. [2] derived the next result in 1991.

###### Theorem 1.9

(Alon, Babai, and Suzuki [2]). Let $\mathcal{L}=\\{l_{1},l_{2},\dots,l_{s}\\}$
be a set of $s$ nonnegative integers. Suppose that $\mathcal{V}$ is a family
of subspaces of $\mathbb{F}_{q}^{n}$ satisfying that $\dim(V_{i}\cap
V_{j})\in\mathcal{L}$ for any distinct subspaces $V_{i}$ and $V_{j}$ in
$\mathcal{V}$ and $\dim(V_{i})\in\\{k_{1},k_{2},\dots,k_{r}\\}$ with
$k_{i}>s-r$ for every $i$. Then

$|\mathcal{V}|\leq\genfrac{[}{]}{0.0pt}{}{n}{s}+\genfrac{[}{]}{0.0pt}{}{n}{s-1}+\cdots+\genfrac{[}{]}{0.0pt}{}{n}{s-r+1}.$

We say that a family $\mathcal{V}$ of subspaces is Sperner (or antichain) if
no subspace is contained in another subspace in $\mathcal{V}$, and
$\mathcal{V}$ is $k$-Sperner if all chains in $\mathcal{V}$ have length at
most $k$. The following well-known Sperner theorem for families of subspaces
of $\mathbb{F}_{q}^{n}$ can be found in [8], which is a vector space analogue
of the classical Sperner theorem.

###### Theorem 1.10

(q-Analogue Sperner Theorem). Assume that $\mathcal{V}$ is a Sperner family of
subspaces of $\mathbb{F}_{q}^{n}$. Then

$|\mathcal{V}|\leq\genfrac{[}{]}{0.0pt}{}{n}{\lfloor\frac{n}{2}\rfloor}.$

We denote $\sum[n,k]$ to be the sum of the $k$ largest $q$-binomial
coefficients of order $n$, i.e.,
$\sum[n,k]=\sum_{i=1}^{k}\left[n\atop\lfloor\frac{n-k}{2}\rfloor+i\right]$.
Let $\sum^{*}[n,k]$ be the collection of families consisting of the
corresponding full levels, i.e., if $n+k$ is odd, then $\sum^{*}[n,k]$
contains one family
$\cup_{i=1}^{k}\left[[n]\atop\lfloor\frac{n-k}{2}\rfloor+i\right]$ (where
$\genfrac{[}{]}{0.0pt}{}{[n]}{k}$ denotes the set of all $k$-dimensional
subspaces of $\mathbb{F}_{q}^{n}$); if $n+k$ is even, then $\sum^{*}[n,k]$
contains two families of the same size
$\cup_{i=0}^{k-1}\left[[n]\atop\frac{n-k}{2}+i\right]$ and
$\cup_{i=1}^{k}\left[[n]\atop\frac{n-k}{2}+i\right]$.

The following $q$-analogue of Erdős’ theorem (Theorem 1.5) is implied by
Theorem 2 in [29], which generalizes Theorem 1.10 to $k$-Sperner families.

###### Theorem 1.11

(Samotij [29]). Assume that $\mathcal{V}$ is a $k$-Sperner family of subspaces
of $\mathbb{F}_{q}^{n}$. Then

$|\mathcal{V}|\leq\sum[n,k].$

Moreover, if $|\mathcal{V}|=\sum[n,k]$, then $\mathcal{V}\in\sum^{*}[n,k]$.

We say that a family $\mathcal{V}$ of subspaces is intersecting if
$\dim(V_{i}\cap V_{j})>0$ for any pair of distinct
$V_{i},V_{j}\in\mathcal{V}$. Here, our first main result is the following
theorem on an intersecting $k$-Sperner family of subspaces of
$\mathbb{F}_{q}^{n}$ which improves considerably the bound in Theorem 1.11,
where $\mathcal{L}_{n,k}[R]$ denotes the set of all $k$-dimensional subspaces
containing a given subspace $R$ in $\mathbb{F}_{q}^{n}$.

###### Theorem 1.12

Suppose that $\mathcal{V}$ is an intersecting $k$-Sperner family of subspaces
of $\mathbb{F}_{q}^{n}$ such that $\dim(V)\leq\lfloor\frac{n}{2}\rfloor$ for
every $V\in\mathcal{V}$. Then

$|\mathcal{V}|\leq\sum_{j=\lfloor\frac{n}{2}\rfloor-k+1}^{\lfloor\frac{n}{2}\rfloor}\genfrac{[}{]}{0.0pt}{}{n-1}{j-1}$

and equality holds if and only if there exists a $1$-dimensional subspace $R$
of $\mathbb{F}_{q}^{n}$ such that $\mathcal{V}_{j}=\\{V\in\mathcal{V}\mid
dim(V)=j\\}=\mathcal{L}_{n,j}[R]$ for each $\lfloor\frac{n}{2}\rfloor-k+1\leq
j\leq\lfloor\frac{n}{2}\rfloor$.

Note that for $\genfrac{[}{]}{0.0pt}{}{n}{k}$ defined by (1), one has

$\genfrac{[}{]}{0.0pt}{}{n}{k}=q^{k}\genfrac{[}{]}{0.0pt}{}{n-1}{k}+\genfrac{[}{]}{0.0pt}{}{n-1}{k-1}.$

The bound in Theorem 1.12 is considerably smaller than that in Theorem 1.11.

We say that a property $P$ of families is hereditary if for any family
$\mathcal{F}$ with property $P$, every subfamily of $\mathcal{F}$ has property
$P$. Clearly, $\mathcal{L}$-intersecting, Sperner, and $k$-Sperner are
hereditary properties.

Our next main result establishes a general relationship between upper bounds
for the sizes of $\mathcal{L}$-intersecting families of subsets of $[n]$ and
upper bounds for the sizes of $\mathcal{L}$-intersecting families of subspaces
of $\mathbb{F}_{q}^{n}$. Denote

$f(n,c_{0},\dots,c_{k})=c_{k}{{n}\choose{k}}+c_{k-1}{{n}\choose{k-1}}+\cdots+c_{0}{{n}\choose{0}},$
$f_{q}(n,c_{0},\dots,c_{k})=c_{k}\genfrac{[}{]}{0.0pt}{}{n}{k}+c_{k-1}\genfrac{[}{]}{0.0pt}{}{n}{k-1}+\cdots+c_{0}\genfrac{[}{]}{0.0pt}{}{n}{0},$

where $0\leq c_{i}\leq 1$ for each $0\leq i\leq k-1$ and $0<c_{k}\leq 1$.

###### Theorem 1.13

Let $P$ be a hereditary property which is $\mathcal{L}$-intersecting or
$T$-configuration forbidden, where $\mathcal{L}$ is a set of nonnegative
integers and $T$ is a given configuration. Assume that $k$ is fixed and $n\geq
2k$. If any family in $2^{[n]}$ satisfying property $P$ has at most
$f(n,c_{0},\dots,c_{k})$ members, then any family of subspaces of
$\mathbb{F}_{q}^{n}$ with property $P$ has at most
$f_{q}(n,c_{0},\dots,c_{k})$ members for $n$ sufficiently large.

By Theorem 1.13, one sees easily that for $n$ sufficiently large, Theorems
1.2–1.4 give rise to Theorems 1.7–1.9, respectively.

Let $0<c\leq 1$ and

$f^{*}(n,k,c)=c{{n}\choose{k}},$
$f^{*}_{q}(n,k,c)=c\genfrac{[}{]}{0.0pt}{}{n}{k}.$

For uniform families, we have the following similar result without requiring
$n$ to be sufficiently large.

###### Theorem 1.14

Let $P$ be a hereditary property which is $\mathcal{L}$-intersecting or
$T$-configuration forbidden, where $\mathcal{L}$ is a set of nonnegative
integers and $T$ is a given configuration. Assume that $k$ is fixed and $n\geq
2k$. If any family in ${{[n]}\choose{k}}$ satisfying property $P$ has at most
$f^{*}(n,k,c)$ members, then any family of $k$-dimensional subspaces of
$\mathbb{F}_{q}^{n}$ with property $P$ has at most $f^{*}_{q}(n,k,c)$ members.

A collection $M$ of pairwise disjoint sets is called a matching. The matching
number $\nu(\mathcal{F})$ of a family $\mathcal{F}\subseteq{{[n]}\choose{k}}$
is the size of the largest matching that $\mathcal{F}$ contains. Note that a
family $\mathcal{F}$ is intersecting if $\nu(\mathcal{F})=1$. Erdős [10]
provided the following well-known conjecture about the maximum size of a
family $\mathcal{F}\subseteq{{[n]}\choose{k}}$ with $\nu(\mathcal{F})\leq s$.

###### Conjecture 1.15

(Erdős Matching Conjecture [10]). Let $\mathcal{F}\subseteq{{[n]}\choose{k}}$
with $n\geq(s+1)k$ and $\nu(\mathcal{F})\leq s$. Then

$|\mathcal{F}|\leq\max\bigg{\\{}{{(s+1)k-1}\choose{k}},\mbox{
}{{n}\choose{k}}-{{n-s}\choose{k}}\bigg{\\}}.$

In section 4, we provide applications of Theorems 1.13 and 1.14. In
particular, we derive generalizations of the well known results about Erdős-
Chvátal simplex conjecture and Erdős matching conjecture.

##  2 Proof of Theorem 1.12

First, we show that $\left[n\atop k\right]$ satisfies the following symmetric
and unimodal properties similar to those of the binomial coefficients.

###### Lemma 2.1

For $0\leq k<l\leq\lfloor\frac{n}{2}\rfloor$, we have

${\rm(i)}\mbox{
}\genfrac{[}{]}{0.0pt}{}{n}{k}=\genfrac{[}{]}{0.0pt}{}{n}{n-k},\hskip
341.43306pt$ ${\rm(ii)}\mbox{
}\genfrac{[}{]}{0.0pt}{}{n}{k}<\genfrac{[}{]}{0.0pt}{}{n}{l}.\hskip
358.50471pt$

Proof. For (i), it follows from the fact

$\genfrac{[}{]}{0.0pt}{}{n}{n-k}=\prod_{0\leq i\leq
n-k-1}\frac{q^{n-i}-1}{q^{n-k-i}-1}\hskip 244.6937pt$
$=\frac{(q^{n}-1)(q^{n-1}-1)\cdots(q^{n-(n-k)+1}-1)}{(q^{n-k}-1)(q^{n-k-1}-1)\cdots(q-1)}\hskip
119.50157pt$ $\hskip
28.45274pt=\frac{(q^{n}-1)(q^{n-1}-1)\cdots(q^{n-(n-k)+1}-1)}{(q^{n-k}-1)(q^{n-k-1}-1)\cdots(q-1)}\cdot\frac{(q^{n}-1)(q^{n-1}-1)\cdots(q-1)}{(q^{n}-1)(q^{n-1}-1)\cdots(q-1)}$
$=\frac{(q^{n}-1)(q^{n-1}-1)\cdots(q^{n-k+1}-1)}{(q^{k}-1)(q^{k-1}-1)\cdots(q-1)}=\genfrac{[}{]}{0.0pt}{}{n}{k}.\hskip
99.58464pt$

For (ii), it suffices to show that for $k+1\leq\lfloor\frac{n}{2}\rfloor$,

$\genfrac{[}{]}{0.0pt}{}{n}{k}<\genfrac{[}{]}{0.0pt}{}{n}{k+1}.$

Note that

$\genfrac{[}{]}{0.0pt}{}{n}{k+1}=\frac{q^{n-k}-1}{q^{k+1}-1}\genfrac{[}{]}{0.0pt}{}{n}{k}.$

For $k+1\leq\lfloor\frac{n}{2}\rfloor$, we have $n-k>k+1$. It follows that

$\hskip
176.40707pt\genfrac{[}{]}{0.0pt}{}{n}{k}<\genfrac{[}{]}{0.0pt}{}{n}{k+1}.\hskip
176.40707pt\hfill\Box$

The following LYM type inequality for intersecting Sperner families of
subspaces of $\mathbb{F}_{q}^{n}$ is given in [31]. Recall that
$\mathcal{L}_{n,k}[R]$ denotes the set of all $k$-dimensional subspaces
containing a given subspace $R$ in $\mathbb{F}_{q}^{n}$.

###### Theorem 2.2

(Wang, [31]). Suppose that $\mathcal{V}$ is an intersecting Sperner family of
subspaces of $\mathbb{F}_{q}^{n}$. Then

$\sum_{k=1}^{\lfloor\frac{n}{2}\rfloor}\frac{|\mathcal{V}_{k}|}{\genfrac{[}{]}{0.0pt}{}{n-1}{k-1}}\leq
1,$

where $\mathcal{V}_{k}=\\{V\in\mathcal{V}\mid dim(V)=k\\}$ for $1\leq
k\leq\lfloor\frac{n}{2}\rfloor$. Equality holds if and only if
$\mathcal{V}=\mathcal{L}_{n,k}[R]$ for some $1\leq
k\leq\lfloor\frac{n}{2}\rfloor$ and some $1$-dimensional subspace $R$ of
$\mathbb{F}_{q}^{n}$.

###### Lemma 2.3

If a family $\mathcal{V}$ of subspaces of $\mathbb{F}_{q}^{n}$ is intersecting
k-Sperner, then $\mathcal{V}$ is the union of $k$ disjoint intersecting
antichains.

Proof. Let $\mathcal{V}=\\{V_{1},V_{2},\dots,V_{m}\\}$ be a family of
subspaces of $\mathbb{F}_{q}^{n}$. Suppose that $\mathcal{V}$ is intersecting
$k$-Sperner. We now show that $\mathcal{V}$ is the union of $k$ intersecting
antichains. Define the required $k$ antichains recursively as follows: Let
$\mathcal{W}_{1}$ denote the family of all minimal subspaces in $\mathcal{V}$,
and if $\mathcal{W}_{j}$ is defined for all $1\leq j<i$, then denote
$\mathcal{W}_{i}$ to be the family of all minimal subspaces in
$\mathcal{V}\setminus(\cup_{j=1}^{i-1}\mathcal{W}_{j})$. This process
continues and stops at $i=h$. Then $\mathcal{W}_{1}$, $\mathcal{W}_{2}$, …,
$\mathcal{W}_{h}$ are disjoint intersecting antichains by definition. We claim
that $h\leq k$. For every $V\in\mathcal{W}_{i}$, there exists a
$V^{\prime}\in\mathcal{W}_{i-1}$ such that $V^{\prime}\subseteq V$. Hence, if
there exists a subspace in $\mathcal{W}_{k+1}$, then we would have a
$(k+1)$-chain in $\mathcal{V}$, a contradiction. Thus, $h\leq k$ and
$\mathcal{V}$ is the union of $k$ disjoint intersecting antichains.
$\hfill\Box$

As an easy consequence of Theorem 2.2 and Lemma 2.3, one has the next
extension of Theorem 2.2 to intersecting $k$-Sperner families.

###### Theorem 2.4

Suppose that $\mathcal{V}$ is an intersecting $k$-Sperner family of subspaces
of $\mathbb{F}_{q}^{n}$. For $0\leq j\leq n$, let
$\mathcal{V}_{j}=\\{V\in\mathcal{V}\mid\dim(V)=j\\}$. If
$\dim(V)\leq\lfloor\frac{n}{2}\rfloor$ for each $V\in\mathcal{V}$, then

$\sum_{j=1}^{\lfloor\frac{n}{2}\rfloor}\frac{|\mathcal{V}_{j}|}{\genfrac{[}{]}{0.0pt}{}{n-1}{j-1}}\leq
k.$

Proof. Suppose that $\mathcal{V}=\\{V_{1},V_{2},\dots,V_{m}\\}$ is an
intersecting $k$-Sperner family of $\mathbb{F}_{q}^{n}$. By Lemma 2.3,
$\mathcal{V}$ is the union of $k$ intersecting antichains $\mathcal{W}_{1}$,
$\mathcal{W}_{2}$, …, $\mathcal{W}_{k}$. Applying Theorem 2.2 on each of these
antichains $\mathcal{W}_{i}$ and adding all the inequalities up, one obtains
the desired result. $\hfill\Box$

###### Lemma 2.5

Suppose that for integers $1\leq k\leq\lfloor\frac{n}{2}\rfloor$ and
nonnegative real numbers $f_{1},f_{2},\dots,f_{\lfloor\frac{n}{2}\rfloor}$,
the following inequalities hold:

$\sum_{j=1}^{\lfloor\frac{n}{2}\rfloor}\frac{|f_{j}|}{\genfrac{[}{]}{0.0pt}{}{n-1}{j-1}}\leq
k,$ $f_{i}\leq\genfrac{[}{]}{0.0pt}{}{n-1}{i-1}\mbox{ for each }1\leq
i\leq\bigg{\lfloor}\frac{n}{2}\bigg{\rfloor}.$

Then

$\sum_{j=1}^{\lfloor\frac{n}{2}\rfloor}f_{j}\leq\sum_{j=\lfloor\frac{n}{2}\rfloor-k+1}^{\lfloor\frac{n}{2}\rfloor}\genfrac{[}{]}{0.0pt}{}{n-1}{j-1}.$

Proof. Consider the vector
$f=(f_{1},f_{2},\dots,f_{\lfloor\frac{n}{2}\rfloor})$ which maximizes $\sum
f_{i}$. For
$\genfrac{[}{]}{0.0pt}{}{n-1}{j-1}<\genfrac{[}{]}{0.0pt}{}{n-1}{i-1}$, the
inequalities $f_{i}<\genfrac{[}{]}{0.0pt}{}{n-1}{i-1}$ and $f_{j}>0$ would
lead to a bigger $\sum f_{i}$ by replacing $f_{i}$ and $f_{j}$ by
$f_{i}+\varepsilon\genfrac{[}{]}{0.0pt}{}{n-1}{i-1}$ and
$f_{j}-\varepsilon\genfrac{[}{]}{0.0pt}{}{n-1}{j-1}$, where $\varepsilon>0$ is
small enough, giving a contradiction. It follows that we have either
$f_{i}=\genfrac{[}{]}{0.0pt}{}{n-1}{i-1}$ or $f_{i}=0$ for each $0\leq
i\leq\lfloor\frac{n}{2}\rfloor$. Thus, by Lemma 2.1, the lemma follows.
$\hfill\Box$

Proof of Theorem 1.12. Suppose that $\mathcal{V}$ is an intersecting
$k$-Sperner family of subspaces of $\mathbb{F}_{q}^{n}$ such that
$\dim(V)\leq\lfloor\frac{n}{2}\rfloor$ for each $V\in\mathcal{V}$. Let
$\mathcal{V}_{j}=\\{V\in\mathcal{V}\mid\dim(V)=j\\}$ and
$f_{j}=|\mathcal{V}_{j}|$ for each $0\leq j\leq\lfloor\frac{n}{2}\rfloor$.
Since $\mathcal{V}$ is intersecting, $\mathcal{V}_{j}$ is intersecting and
$j$-uniform for each $1\leq j\leq\lfloor\frac{n}{2}\rfloor$. It follows from
Theorem 1.6 that
$f_{j}=|\mathcal{V}_{j}|\leq\genfrac{[}{]}{0.0pt}{}{n-1}{j-1}$ for every
$1\leq j\leq\lfloor\frac{n}{2}\rfloor$. By Theorem 2.4, we have

$\sum_{j=1}^{\lfloor\frac{n}{2}\rfloor}\frac{|\mathcal{V}_{j}|}{\genfrac{[}{]}{0.0pt}{}{n-1}{j-1}}\leq
k.$ (2)

It follows from Lemma 2.5 that

$|\mathcal{V}|=\sum_{j=1}^{\lfloor\frac{n}{2}\rfloor}f_{j}\leq\sum_{j=\lfloor\frac{n}{2}\rfloor-k+1}^{\lfloor\frac{n}{2}\rfloor}\genfrac{[}{]}{0.0pt}{}{n-1}{j-1}.$

Suppose that the equality holds above. Then it follows from the proof of Lemma
2.5 that we must have
$|\mathcal{V}_{i}|=f_{i}=\genfrac{[}{]}{0.0pt}{}{n-1}{i-1}$ or
$|\mathcal{V}_{i}|=f_{i}=0$ for each $0\leq i\leq\lfloor\frac{n}{2}\rfloor$.
Thus, it follows from Theorem 1.6 that
$\mathcal{V}_{j}=\mathcal{L}_{n,j}[R_{j}]$ with $dim(R_{j})=1$ for each
$\lfloor\frac{n}{2}\rfloor-k+1\leq j\leq\lfloor\frac{n}{2}\rfloor$. Moreover,
since $n\geq 2k+1$ and $\mathcal{V}$ is intersecting, we must have
$R_{i}=R_{j}=R$ for all $\lfloor\frac{n}{2}\rfloor-k+1\leq
i,j\leq\lfloor\frac{n}{2}\rfloor$, where $R$ is a fixed $1$-dimensional
subspace of $\mathbb{F}_{q}^{n}$. For otherwise, if $R_{i}\neq R_{j}$ for some
$\lfloor\frac{n}{2}\rfloor-k+1\leq i,j\leq\lfloor\frac{n}{2}\rfloor$ with
$i\neq j$, then there exist $W_{i}\in\mathcal{V}_{i}$ and
$W_{j}\in\mathcal{V}_{j}$ such that $dim(W_{i}\cap W_{j})=0$, contradicting
the assumption that $\mathcal{V}$ is intersecting. $\hfill\Box$

##  3 Proofs of Theorems 1.13 and 1.14

We begin with the following lemma.

###### Lemma 3.1

Let $k\geq 1$ be fixed and $n\geq 2k+2$. For any $\varepsilon>0$, there exists
$n_{0}$ such that for $n\geq n_{0}$, we have

${\rm(i)}\mbox{
}\sum_{i=0}^{k}{{n}\choose{i}}<\varepsilon{{n}\choose{k+1}},\hskip 312.9803pt$
${\rm(ii)}\mbox{
}\sum_{i=0}^{k}\genfrac{[}{]}{0.0pt}{}{n}{i}<\varepsilon\genfrac{[}{]}{0.0pt}{}{n}{k+1}.\hskip
312.9803pt$

Proof. We first prove (i). Since
${{n}\choose{k+1}}=\frac{n-k}{k+1}{{n}\choose{k}}$, we have

$\varepsilon{{n}\choose{k+1}}-(k+1){{n}\choose{k}}=\bigg{(}\frac{\varepsilon(n-k)}{k+1}-(k+1)\bigg{)}{{n}\choose{k}}.$

For $n>\frac{(k+1)^{2}}{\varepsilon}+k$, we have
$\frac{\varepsilon(n-k)}{k+1}-(k+1)>0$. Note that
$\sum_{i=0}^{k}{{n}\choose{i}}\leq(k+1){{n}\choose{k}}$. It follows that

$\varepsilon{{n}\choose{k+1}}>(k+1){{n}\choose{k}}\geq\sum_{i=0}^{k}{{n}\choose{i}}.$

For (ii), note that

$\genfrac{[}{]}{0.0pt}{}{n}{k+1}=\frac{q^{n-k}-1}{q^{k+1}-1}\genfrac{[}{]}{0.0pt}{}{n}{k}.$

Similar to (i), by applying Lemma 2.1, one has the desired inequality in (ii).
$\hfill\Box$

As a consequence of Lemma 3.1, one can derive the next fact easily. Recall
that

$f_{q}(n,c_{0},\dots,c_{k})=c_{k}\genfrac{[}{]}{0.0pt}{}{n}{k}+c_{k-1}\genfrac{[}{]}{0.0pt}{}{n}{k-1}+\cdots+c_{0}\genfrac{[}{]}{0.0pt}{}{n}{0},$

where $0\leq c_{i}\leq 1$ for each $0\leq i\leq k$.

###### Lemma 3.2

Let $k\geq 1$ be fixed and $n\geq 2k$. For $n$ sufficiently large, if
$f_{q}(n,b_{0},\dots,b_{k})>f_{q}(n,c_{0},\dots,c_{k})$, then we have
$b_{k}>c_{k}$ or there exists $j_{0}<k$ such that $b_{j}=c_{j}$ for all
$j_{0}+1\leq j\leq k$ and $b_{j_{0}}>c_{j_{0}}$.

Proof. Suppose that $j_{0}$ is the largest integer between $0$ and $k$ such
that $b_{j_{0}}\neq c_{j_{0}}$. Then we have either $b_{k}\neq c_{k}$ or
$j_{0}<k$ and $b_{j}=c_{j}$ for all $j_{0}+1\leq j\leq k$. By Lemma 3.1 (ii),
we must have either $b_{k}>c_{k}$ or $b_{j}=c_{j}$ for all $j_{0}+1\leq j\leq
k$ and $b_{j_{0}}>c_{j_{0}}$ for $n$ sufficiently large. $\hfill\Box$

Next, we introduce some notions and the following covering lemma by Gerbner
[15]. Let $S$ be a set and $S=S_{0}\cup S_{1}\cup\cdots\cup S_{n}$ be a
partition (In the discussions here, $S$ is either the set $2^{[n]}$ of all
subsets of $[n]$ or the set of all subspaces of $\mathbb{F}_{q}^{n}$, and
$S_{i}$ will be level $i$, i.e., $S_{i}={{[n]}\choose{i}}$ or the set of all
$i$-dimensional subspaces of $\mathbb{F}_{q}^{n}$, respectively). Given a
vector $t=(t_{0},t_{1},\dots,t_{n})$, we say a family $\Gamma$ of subsets of
$S$ is a $t$-covering family of $S$ if for each $0\leq i\leq n$, each member
in $S_{i}$ is contained in exactly $t_{i}$ sets in the family $\Gamma$.

Given a family $\mathcal{F}\subseteq S$, let $f_{i}=|\mathcal{F}\cap S_{i}|$
and $(f_{0},f_{1},\dots,f_{n})$ is called the profile vector of $\mathcal{F}$.
For a weight $\overline{w}=(w_{0},w_{1},\dots,w_{n})$ and a family
$\mathcal{F}\subseteq S$, let
$\overline{w}(\mathcal{F})=\sum_{i=0}^{n}w_{i}|\mathcal{F}\cap S_{i}|$. Denote
$\overline{w/t}=(w_{0}/t_{0},w_{1}/t_{1},\dots,w_{n}/t_{n})$. The following
lemma is Lemma 2.1 in [15].

###### Lemma 3.3

Let $P$ be a hereditary property of subsets (families) of $S$ and $\Gamma$ be
a $t$-covering family of $S$. Assume that there exists a real number $x$ such
that for every $G\in\Gamma$, every subset $G^{\prime}$ of $G$ with property
$P$ has $\overline{w/t}(G^{\prime})\leq x$. Then
$\overline{w}(F)\leq|\Gamma|x$ for every $F\subseteq S$ with property $P$.

Let $S=\mathcal{L}_{n}(q)$ be the set of all subspaces of
$\mathbb{F}_{q}^{n}$. Denote $S=S_{0}\cup S_{1}\cup\cdots\cup S_{n}$ for a
partition of $S$ such that $S_{i}$ is the set of all $i$-dimensional subspaces
of $\mathbb{F}_{q}^{n}$. We construct a $t$-covering family $\Gamma$ of $S$
such that every $\mathcal{G}\in\Gamma$ is a subfamily in $\mathcal{L}_{n}(q)$
isomorphic to Boolean lattice $\mathcal{B}_{n}$ (the set $2^{[n]}$ of all
subsets of $[n]$ with the ordering being the containment relation) as follows:
Choose an arbitrary basis $B=\\{v_{1},\dots,v_{n}\\}$ of $\mathbb{F}_{q}^{n}$
and let $\mathcal{G}_{B}=\\{span(U)\mid U\subseteq B\\}$, i.e., the family of
all subspaces that are generated by subsets of the vectors in $B$. Obviously
the function that maps $H\subseteq[n]$ to the subspace $span\\{v_{x}\mid x\in
H\\}$ keeps inclusion and intersection properties. Let $\Gamma$ be the
collection of the families $\mathcal{G}_{B}$ over all bases $B$, i.e.,
$\Gamma=\\{\mathcal{G}_{B}\mid B\mbox{ is a basis of }\mathbb{F}_{q}^{n}\\}$.
Denote

$\alpha(q,n)=\frac{(q^{n}-1)(q^{n}-q)(q^{n}-q^{2})\cdots(q^{n}-q^{n-1})}{n!}.$
(3)

Let $t=(t_{0},t_{1},\dots,t_{n})$ be such that for $0\leq i\leq n$,

$t_{i}=\frac{(q^{i}-1)(q^{i}-q)\cdots(q^{i}-q^{i-1})(q^{n}-q^{i})\cdots(q^{n}-q^{n-1})}{i!(n-i)!}.$
(4)

The next lemma follows from an easy counting.

###### Lemma 3.4

Let $S=\mathcal{L}_{n}(q)$ be the set of all subspaces of $\mathbb{F}_{q}^{n}$
and $\Gamma=\\{\mathcal{G}_{B}\mid B\mbox{ is a basis of
}\mathbb{F}_{q}^{n}\\}$. Then $|\Gamma|=\alpha(q,n)$ and $\Gamma$ is a
$t$-covering of $S$ with $t=(t_{0},t_{1},\dots,t_{n})$ given in (4).

Proof. To show $|\Gamma|=\alpha(q,n)$, it suffices to show that
$\mathbb{F}_{q}^{n}$ has $\alpha(q,n)$ different bases. To obtain a basis
$B=\\{v_{1},v_{2},\dots,v_{n}\\}$ for $\mathbb{F}_{q}^{n}$, one can choose $n$
nonzero linearly independent vectors in the order $v_{1},v_{2},\dots,v_{n}$ so
that $v_{j}\not\in span\\{v_{1},v_{2},\dots,v_{j-1}\\}$ as follows: There are
$q^{n}-1$ ways to choose $v_{1}$, and then there are $q^{n}-q$ ways to choose
$v_{2}$, $q^{n}-q^{2}$ ways to choose $v_{3}$, …, $q^{n}-q^{n-1}$ ways to
choose $v_{n}$. Thus, there are
$(q^{n}-1)(q^{n}-q)(q^{n}-q^{2})\cdots(q^{n}-q^{n-1})$ combined ways to form
$B=\\{v_{1},v_{2},\dots,v_{n}\\}$ (in order). Clearly, there are $n!$ ways
(permutations on $B$) to form the same $B$ (without order). It follows that
there are

$\alpha(q,n)=\frac{(q^{n}-1)(q^{n}-q)(q^{n}-q^{2})\cdots(q^{n}-q^{n-1})}{n!}$

different bases in $\mathbb{F}_{q}^{n}$.

For any $i$-dimensional subspace $V$ in $\mathbb{F}_{q}^{n}$, similar to the
argument above, there are

$\frac{(q^{i}-1)(q^{i}-q)\cdots(q^{i}-q^{i-1})}{i!}$

different bases in $V$. For each basis $D$ in $V$, one can extend $D$ to a
basis of $\mathbb{F}_{q}^{n}$ in

$\frac{(q^{n}-q^{i})(q^{n}-q^{i+1})\cdots(q^{n}-q^{n-1})}{(n-i)!}$

different ways. Therefore, $V$ is contained in

$t_{i}=\frac{(q^{i}-1)(q^{i}-q)\cdots(q^{i}-q^{i-1})(q^{n}-q^{i})\cdots(q^{n}-q^{n-1})}{i!(n-i)!}$

different $\mathcal{G}_{B}\in\Gamma$. It follows that $\Gamma$ is a
$t$-covering of $S=\mathcal{L}_{n}(q)$. $\hfill\Box$

The following proof for Theorem 1.13 is motivated by the proof of Theorem 1.3
in [15].

Proof of Theorem 1.13. Assume that $\mathcal{V}$ is a family in
$\mathcal{L}_{n}(q)$ satisfying property $P$ which is either
$\mathcal{L}$-intersecting or $T$-configuration forbidden. Let
$\Gamma=\\{\mathcal{G}_{B}\mid B\mbox{ is a basis of }\mathbb{F}_{q}^{n}\\}$.
By Lemma 3.4, $\Gamma$ is a $t$-covering of $S=\mathcal{L}_{n}(q)$. Let
$w_{i}=t_{i}$ for each $0\leq i\leq n$. Then $\overline{w/t}=(1,1,\dots,1)$.
Since every $\mathcal{G}\in\Gamma$ is isomorphic to Boolean lattice
$\mathcal{B}_{n}$, by the assumption, the largest weight
$\overline{w/t}(\mathcal{G}^{\prime})$ among all subfamilies
$\mathcal{G}^{\prime}\subseteq\mathcal{G}$ satisfying $P$, i.e., the largest
cardinality of such $\mathcal{G}^{\prime}\subseteq\mathcal{G}$ is
$f(n,c_{0},\dots,c_{k})$. By Lemma 3.3, we have

$\overline{w}(\mathcal{V})\leq|\Gamma|f(n,c_{0},\dots,c_{k}).$ (5)

To the contrary, suppose that $|\mathcal{V}|>f_{q}(n,c_{0},\dots,c_{k})$. Then
there exists $b=(b_{0},b_{1},\dots,b_{n})$ with $0\leq b_{j}\leq 1$ for $0\leq
j\leq n$ such that

$|\mathcal{V}|=g_{q}(n,b_{0},\dots,b_{n})=b_{n}\genfrac{[}{]}{0.0pt}{}{n}{n}+b_{n-1}\genfrac{[}{]}{0.0pt}{}{n}{n-1}+\cdots+b_{0}\genfrac{[}{]}{0.0pt}{}{n}{0}$

with

$\bigg{|}\mathcal{V}\cap\genfrac{[}{]}{0.0pt}{}{[n]}{j}\bigg{|}=b_{j}\genfrac{[}{]}{0.0pt}{}{n}{j}\mbox{
for }0\leq j\leq n,$

where $\genfrac{[}{]}{0.0pt}{}{[n]}{j}$ denotes the set of all $j$-dimensional
subspaces of $\mathbb{F}_{q}^{n}$. It follows that

$\overline{w}(\mathcal{V})=\sum_{i=0}^{n}b_{i}w_{i}\genfrac{[}{]}{0.0pt}{}{n}{i}\hskip
270.30118pt$
$=\sum_{i=0}^{n}b_{i}\frac{(q^{i}-1)(q^{i}-q)\cdots(q^{i}-q^{i-1})(q^{n}-q^{i})\cdots(q^{n}-q^{n-1})}{i!(n-i)!}\genfrac{[}{]}{0.0pt}{}{n}{i}$
$=\sum_{i=0}^{n}b_{i}\frac{(q^{i}-1)\cdots(q^{i}-q^{i-1})(q^{n}-q^{i})\cdots(q^{n}-q^{n-1})}{i!(n-i)!}\hskip
51.21495pt$ $\hskip
187.78818pt\cdot\frac{(q^{n}-1)(q^{n}-q)\cdots(q^{n}-q^{n-1})}{(q^{i}-1)\cdots(q^{i}-q^{i-1})(q^{n}-q^{i})\cdots(q^{n}-q^{n-1})}$
$=\sum_{i=0}^{n}b_{i}\frac{\alpha(n,q)n!}{i!(n-i)!}=\sum_{i=0}^{h}b_{i}|\Gamma|{{n}\choose{i}}\hskip
145.10905pt$ $=g(n,b_{0},\dots,b_{n})|\Gamma|,\hskip 204.85983pt$

where $g(n,b_{0},\dots,b_{n})=\sum_{i=0}^{n}b_{i}{{n}\choose{i}}$. Combining
with (5), we obtain

$g(n,b_{0},\dots,b_{n})\leq f(n,c_{0},\dots,c_{k}).$ (6)

Since ${{n}\choose{j}}={{n}\choose{n-j}}$ and
$\genfrac{[}{]}{0.0pt}{}{n}{j}=\genfrac{[}{]}{0.0pt}{}{n}{n-j}$ for $0\leq
j\leq\lfloor\frac{n}{2}\rfloor$ by Lemma 2.1, we may combine symmetric $j$-th
and $(n-j)$-th terms in $g_{q}(n,b_{0},\dots,b_{n})$ and
$g(n,b_{0},\dots,b_{n})=\sum_{i=0}^{n}b_{i}{{n}\choose{i}}$ for $0\leq
j\leq\lfloor\frac{n}{2}\rfloor$, and obtain

$|\mathcal{V}|=g_{q}(n,b_{0},\dots,b_{n})=g_{q}(n,b^{\prime}_{0},\dots,b^{\prime}_{\lfloor\frac{n}{2}\rfloor})=b^{\prime}_{\lfloor\frac{n}{2}\rfloor}\genfrac{[}{]}{0.0pt}{}{n}{\lfloor\frac{n}{2}\rfloor}+b^{\prime}_{\lfloor\frac{n}{2}\rfloor-1}\genfrac{[}{]}{0.0pt}{}{n}{\lfloor\frac{n}{2}\rfloor-1}+\cdots+b^{\prime}_{0}\genfrac{[}{]}{0.0pt}{}{n}{0},$
$g(n,b_{0},\dots,b_{n})=g(n,b^{\prime}_{0},\dots,b^{\prime}_{\lfloor\frac{n}{2}\rfloor})=b^{\prime}_{\lfloor\frac{n}{2}\rfloor}{{n}\choose{\lfloor\frac{n}{2}\rfloor}}+b^{\prime}_{\lfloor\frac{n}{2}\rfloor-1}{{n}\choose{\lfloor\frac{n}{2}\rfloor-1}}+\dots+b^{\prime}_{0}{{n}\choose{0}},$

where $0\leq b_{j}\leq b^{\prime}_{j}=b_{j}+b_{n-j}\leq 2$ for each $0\leq
j\leq\lfloor\frac{n}{2}\rfloor$ with an exception that
$b^{\prime}_{\frac{n}{2}}=b_{\frac{n}{2}}$ when $n$ is even. Assume that $h$
is the largest integer such that $h\leq\lfloor\frac{n}{2}\rfloor$ and
$b^{\prime}_{h}>0$. Then we have

$|\mathcal{V}|=g_{q}(n,b_{0},\dots,b_{n})=g_{q}(n,b^{\prime}_{0},\dots,b^{\prime}_{h})>f_{q}(n,c_{0},\dots,c_{k}).$

By Lemma 3.1 (ii), we must have either $h>k$ or $h=k$ and

$|\mathcal{V}|=g_{q}(n,b^{\prime}_{0},\dots,b^{\prime}_{k})>f_{q}(n,c_{0},\dots,c_{k}).$

We consider the following two cases:

Case 1. $h=k$. We have
$g_{q}(n,b^{\prime}_{0},\dots,b^{\prime}_{k})=f_{q}(n,b^{\prime}_{0},\dots,b^{\prime}_{k})>f_{q}(n,c_{0},\dots,c_{k})$.
It follows from Lemma 3.2 that either $b^{\prime}_{k}>c_{k}$ or there exists
$j_{0}<k$ such that $b^{\prime}_{j}=c_{j}$ for all $j_{0}+1\leq j\leq k$ and
$b^{\prime}_{j_{0}}>c_{j_{0}}$. Thus, by Lemma 3.1 (i), we have that for $n$
sufficiently large,

$g(n,b_{0},\dots,b_{n})=g(n,b^{\prime}_{0},\dots,b^{\prime}_{k})=\sum_{i=0}^{k}b^{\prime}_{i}{{n}\choose{i}}=f(n,b^{\prime}_{0},\dots,b^{\prime}_{k})>f(n,c_{0},\dots,c_{k}),$

contradicting (6).

Case 2. $h>k$. Since $k+1\leq h\leq\lfloor\frac{n}{2}\rfloor$, we have
${{n}\choose{h}}\geq{{n}\choose{k+1}}$. It follows from Lemma 3.1 (i) that

$g(n,b_{0},\dots,b_{n})=g(n,b^{\prime}_{0},\dots,b^{\prime}_{h})\geq
b^{\prime}_{h}{{n}\choose{k+1}}>f(n,c_{0},\dots,c_{k})$

for $n$ sufficiently large, contradicting (6) again.

Therefore, we have

$|\mathcal{V}|\leq f_{q}(n,c_{0},\dots,c_{k}).$

$\hfill\Box$

We remark that the inequality (6) in the proof of Theorem 1.13 above can be
obtained without using Lemma 3.3 as follows: By Lemma 3.4,
$\Gamma=\\{\mathcal{G}_{B}\mid B\mbox{ is a basis of }\mathbb{F}_{q}^{n}\\}$
is a $t$-covering of $S=\mathcal{L}_{n}(q)$, i.e., every $i$-dimensional
subspace in $S_{i}=\genfrac{[}{]}{0.0pt}{}{[n]}{i}$ is contained in $t_{i}$
members in $\Gamma$. Denote
$\mathcal{V}_{j}=\mathcal{V}\cap\genfrac{[}{]}{0.0pt}{}{[n]}{j}$ for each
$0\leq j\leq n$. Let

$|\mathcal{V}|=g_{q}(n,b_{0},\dots,b_{n})=b_{n}\genfrac{[}{]}{0.0pt}{}{n}{n}+b_{n-1}\genfrac{[}{]}{0.0pt}{}{n}{n-1}+\cdots+b_{0}\genfrac{[}{]}{0.0pt}{}{n}{0}$

with

$|\mathcal{V}_{j}|=\bigg{|}\mathcal{V}\cap\genfrac{[}{]}{0.0pt}{}{[n]}{j}\bigg{|}=b_{j}\genfrac{[}{]}{0.0pt}{}{n}{j}\mbox{
for }0\leq j\leq n,$

where $0\leq b_{j}\leq 1$ for every $0\leq j\leq n$. Since $\mathcal{G}_{B}$
is isomorphic to Boolean lattice $\mathcal{B}_{n}$ for each basis $B$ of
$\mathbb{F}_{q}^{n}$, we have

$\sum_{\mathcal{G}_{B}\in\Gamma}|\mathcal{V}\cap\mathcal{G}_{B}|\leq|\Gamma|\cdot
f(n,c_{0},\dots,c_{k}).$

On the other hand, since every $i$-dimensional subspace in $S_{i}$ is
contained in $t_{i}$ members in $\Gamma$, it follows that

$\sum_{\mathcal{G}_{B}\in\Gamma}|\mathcal{V}\cap\mathcal{G}_{B}|=\sum_{\mathcal{G}_{B}\in\Gamma}\sum_{i=0}^{n}|\mathcal{V}_{i}\cap\mathcal{G}_{B}|=\sum_{i=0}^{n}\sum_{\mathcal{G}_{B}\in\Gamma}|\mathcal{V}_{i}\cap\mathcal{G}_{B}|\hskip
34.1433pt$ $\hskip 54.06023pt=\sum_{i=0}^{n}|\mathcal{V}_{i}|\cdot
t_{i}=\sum_{i=0}^{n}b_{i}\genfrac{[}{]}{0.0pt}{}{n}{i}\cdot
t_{i}=\sum_{i=0}^{n}b_{i}|\Gamma|{{n}\choose{i}},$

where the last equality is due to the following double counting

$\genfrac{[}{]}{0.0pt}{}{n}{i}\cdot
t_{i}=\bigg{|}\\{(V,\mathcal{G}_{B})\mid\dim(V)=i,\mathcal{G}_{B}\in\Gamma\mbox{
and }V\in\mathcal{G}_{B}\\}\bigg{|}=|\Gamma|{{n}\choose{i}}.$

Combining the expressions above, we have

$g(n,b_{0},\dots,b_{n})=\sum_{i=0}^{n}b_{i}{{n}\choose{i}}\leq
f(n,c_{0},\dots,c_{k}).$

Proof of Theorem 1.14. Assume that $\mathcal{V}$ is a family of
$k$-dimensional subspaces of $\mathbb{F}_{q}^{n}$ satisfying property $P$. Let

$|\mathcal{V}|=g_{q}(n,k,a)=a\genfrac{[}{]}{0.0pt}{}{n}{k}\mbox{ with }0<a\leq
1.$

Similar to the proof of Theorem 1.13 above, we have

$g(n,k,a)=a{{n}\choose{k}}\leq f^{*}(n,k,c)=c{{n}\choose{k}}.$ (7)

To the contrary, suppose that

$|\mathcal{V}|=g_{q}(n,k,a)=a\genfrac{[}{]}{0.0pt}{}{n}{k}>f^{*}_{q}(n,k,c)=c\genfrac{[}{]}{0.0pt}{}{n}{k}.$

Then we have $a>c$ which implies

$g(n,k,a)=a{{n}\choose{k}}>f^{*}(n,k,c)=c{{n}\choose{k}},$

contradicting (7). Therefore, the theorem follows. $\hfill\Box$

##  4 Applications

Clearly, by applying Theorem 1.13, one concludes that, for $n$ sufficiently
large, Theorems 1.7–1.9 follow from Theorems 1.2–1.4, respectively.

Next, we provide some applications of Theorem 1.14.

Note that

${{n-1}\choose{k-1}}=\frac{k}{n}{{n}\choose{k}},$ (8)
$\genfrac{[}{]}{0.0pt}{}{n-1}{k-1}=\frac{q^{k}-1}{q^{n}-1}\genfrac{[}{]}{0.0pt}{}{n}{k}.$
(9)

###  4.1 Erdős-Chvátal Simplex Conjecture

A $d$-$simplex$ is defined to be a collection $A_{1},\dots,A_{d+1}$ of subsets
of size $k$ of $[n]$ such that the intersection of all of them is empty, but
the intersection of any $d$ of them is non-empty. A $d$-$cluster$ is a
collection $A_{1},\dots,A_{d+1}$ of subsets of size $k$ of $[n]$ such that
$\cap_{i=1}^{d+1}A_{i}=\emptyset$ and $|\cup_{i=1}^{d+1}A_{i}|\leq 2k$. If
$\\{A_{1},\dots,A_{d+1}\\}$ is both a $d$-simplex and a $d$-cluster, we say
that it is a $d$-simplex-cluster.

In 1971, Erdős conjectured that a family $\mathcal{F}$ of $k$-subsets of $[n]$
with no $2$-simplex (also known as a triangle) satisfies

$|\mathcal{F}|\leq{{n-1}\choose{k-1}}.$

In 1974, Chvátal [5] generalized Erdős’ conjecture as follows.

###### Conjecture 4.1

(Erdős-Chvátal Simplex Conjecture [5]). Suppose $k\geq d+1\geq 3$,
$n\geq\frac{k(d+1)}{d}$, and $\mathcal{F}$ is a family of $k$-subsets of $[n]$
with no $d$-simplex. Then

$|\mathcal{F}|\leq{{n-1}\choose{k-1}}={{n}\choose{k}}-{{n-1}\choose{k}},$

with equality only if $\mathcal{F}$ is a star.

In 2005, Mubayi and Verstraëte [26] proved that the Erdős-Chvátal Simplex
Conjecture is true for $d=2$, and in 2021, Currier [6] proved that the
conjecture holds when $d\geq 3$ and $n\geq 2k-d+2$ as shown below.

###### Theorem 4.2

(Mubayi and Verstraëte [26], Currier [6]). Suppose $k\geq d+1\geq 3$, $n\geq
2k-d+2$, and $\mathcal{F}$ is a family of $k$-subsets of $[n]$ with no
$d$-simplex-cluster. Then

$|\mathcal{F}|\leq{{n-1}\choose{k-1}}={{n}\choose{k}}-{{n-1}\choose{k}},$

with equality if $\mathcal{F}$ is a star.

Let $\mathcal{A}=\\{A_{1},A_{2},\dots,A_{d+1}\\}$ be a family of
$k$-dimensional subspaces of $\mathbb{F}_{q}^{n}$. We say that $\mathcal{A}$
is a $d$-$simplex$ if $\dim(\cap_{i=1}^{d+1}A_{i})=0$ but
$\dim(\cap_{j=1}^{d}A_{i_{j}})>0$ for any $d$ distinct members
$A_{i_{1}},\dots,A_{i_{d}}\in\mathcal{A}$. A family
$\mathcal{A}=\\{A_{1},A_{2},\dots,A_{d+1}\\}$ of subspaces of
$\mathbb{F}^{n}_{q}$ is a $d$-$cluster$ if $\dim(\cap_{i=1}^{d+1}A_{i})=0$ and
$\dim(span(\cup_{i=1}^{d+1}A_{i}))\leq 2k$. A family
$\\{A_{1},\dots,A_{d+1}\\}$ is a $d$-simplex-cluster if it is both a
$d$-simplex and a $d$-cluster.

Recall that

$\genfrac{[}{]}{0.0pt}{}{n}{k}=q^{k}\genfrac{[}{]}{0.0pt}{}{n-1}{k}+\genfrac{[}{]}{0.0pt}{}{n-1}{k-1}=\genfrac{[}{]}{0.0pt}{}{n-1}{k}+q^{n-k}\genfrac{[}{]}{0.0pt}{}{n-1}{k-1}.$

The following asymptotic $q$-analogue of Theorem 4.2 follows easily from
Theorems 1.14 and 4.2.

###### Theorem 4.3

Let $k\geq d+1\geq 3$. Suppose that $\mathcal{V}$ is a family of
$k$-dimensional subspaces of $\mathbb{F}_{q}^{n}$ with no $d$-simplex-cluster.
Then for $n$ sufficiently large,

$|\mathcal{V}|<q^{n-k}\genfrac{[}{]}{0.0pt}{}{n-1}{k-1}=\genfrac{[}{]}{0.0pt}{}{n}{k}-\genfrac{[}{]}{0.0pt}{}{n-1}{k}.$

Proof. First, note that forbidding $d$-simplex-cluster is a hereditary
property. Using the identity (8) and applying Theorems 1.14 and 4.2 with

$f^{*}(n,k,\frac{k}{n})=\frac{k}{n}{{n}\choose{k}}={{n-1}\choose{k-1}},$

we conclude that

$|\mathcal{V}|\leq
f^{*}_{q}(n,k,\frac{k}{n})=\frac{k}{n}\genfrac{[}{]}{0.0pt}{}{n}{k}.$ (10)

Since
$\lim_{n\longrightarrow\infty}\frac{k}{n}\frac{q^{n}-1}{(q^{k}-1)q^{n-k}}=0$,
it follows from (9) and (10) that for $n$ sufficiently large,

$|\mathcal{V}|\leq\frac{k}{n}\genfrac{[}{]}{0.0pt}{}{n}{k}=\frac{k}{n}\cdot\frac{q^{n}-1}{q^{k}-1}\genfrac{[}{]}{0.0pt}{}{n-1}{k-1}\hskip
96.73936pt$
$=\frac{k}{n}\frac{q^{n}-1}{(q^{k}-1)q^{n-k}}q^{n-k}\genfrac{[}{]}{0.0pt}{}{n-1}{k-1}<q^{n-k}\genfrac{[}{]}{0.0pt}{}{n-1}{k-1}.$

$\hfill\Box$

Recently, Liu [24] proved the next result which confirms a conjecture by
Mubayi and Verstraëte [26].

###### Theorem 4.4

(Liu [24]). Let $d\geq k\geq 4$ and $n$ be sufficiently large. Suppose that
$\mathcal{F}$ is a family of $k$-subsets of $[n]$ with no non-trivial
intersecting subfamily of size $d+1$. Then

$|\mathcal{F}|\leq{{n-1}\choose{k-1}}={{n}\choose{k}}-{{n-1}\choose{k}},$

with equality if $\mathcal{F}$ is a star.

Since forbidden a non-trivial intersecting subfamily of size $d+1$ is a
hereditary property, the following $q$-analogue of Theorem 4.4 follows from
Theorems 1.14 and 4.4 and (8) and (9).

###### Theorem 4.5

Let $d\geq k\geq 4$ and $n$ be sufficiently large. Suppose that $\mathcal{V}$
is a family of $k$-dimensional subspaces of $\mathbb{F}_{q}^{n}$ with no non-
trivial intersecting subfamily of size $d+1$. Then

$|\mathcal{V}|\leq
q^{n-k}\genfrac{[}{]}{0.0pt}{}{n-1}{k-1}=\genfrac{[}{]}{0.0pt}{}{n}{k}-\genfrac{[}{]}{0.0pt}{}{n-1}{k}.$

###  4.2 Erdős Matching Conjecture

Recall that the matching number $\nu(\mathcal{F})$ of a family
$\mathcal{F}\subseteq{{[n]}\choose{k}}$ is the size of the largest matching
that $\mathcal{F}$ contains and recall Erdős Matching Conjecture (or EMC for
short) from Conjecture 1.15. Clearly, the case $s=1$ of the EMC is the
classical Erdős-Ko-Rado theorem. Bollobás, Daykin and Erdős [3] established
the EMC for $n\geq 2k^{3}s$; Huang, Loh and Sudakov [19] proved the EMC for
$n\geq 3k^{2}s$; and Frankl [12] showed that the EMC holds for
$n\geq(2s+1)k-s$.

###### Theorem 4.6

(Bollobás, Daykin and Erdős [3], Frankl [12], Huang, Loh and Sudakov [19]).
Let $\mathcal{F}\subseteq{{[n]}\choose{k}}$ with $\nu(\mathcal{F})\leq s$ and
$n\geq(2s+1)k-s$. Then

$|\mathcal{F}|\leq{{n}\choose{k}}-{{n-s}\choose{k}}.$

We say that two subspaces are disjoint if their intersection is the trivial
subspace (i.e., $0$-dimensional subspace). A collection of pairwise disjoint
subspaces is called a matching. Similar to the matching number
$\nu(\mathcal{F})$ of $\mathcal{F}\subseteq{{[n]}\choose{k}}$, one can define
the matching number $\nu_{q}(\mathcal{V})$ for a family $\mathcal{V}$ of
$k$-dimensional subspaces of $\mathbb{F}_{q}^{n}$ to be the size of the
largest matching that $\mathcal{V}$ contains.

Observe that a family $\mathcal{F}$ has matching number $\nu(\mathcal{F})\leq
s$ is a hereditary property. By applying Theorem 1.14, we obtain the next
asymptotic $q$-analogue of Theorem 4.6. Note that for $n\geq 2k+s$ and $s\geq
1$, one has

$\genfrac{[}{]}{0.0pt}{}{n}{k}-\genfrac{[}{]}{0.0pt}{}{n-1}{k}\leq\genfrac{[}{]}{0.0pt}{}{n}{k}-\genfrac{[}{]}{0.0pt}{}{n-s}{k}.$

###### Theorem 4.7

Suppose that $\mathcal{V}$ is a family of $k$-dimensional subspaces of
$\mathbb{F}_{q}^{n}$ with $\nu_{q}(\mathcal{V})\leq s$ and $q\geq 2$. Then for
$n$ sufficiently large,

$|\mathcal{V}|\leq
q^{n-k}\genfrac{[}{]}{0.0pt}{}{n-1}{k-1}=\genfrac{[}{]}{0.0pt}{}{n}{k}-\genfrac{[}{]}{0.0pt}{}{n-1}{k}.$

Proof. For $n\geq(2k+1)s$, by the identity (8), we have

${{n}\choose{k}}-{{n-s}\choose{k}}\hskip 284.52756pt$ $\hskip
56.9055pt=\bigg{(}{{n}\choose{k}}-{{n-1}\choose{k}}\bigg{)}+\bigg{(}{{n-1}\choose{k}}-{{n-2}\choose{k}}\bigg{)}+\cdots+\bigg{(}{{n-s+1}\choose{k}}-{{n-s}\choose{k}}\bigg{)}$
$={{n-1}\choose{k-1}}+{{n-2}\choose{k-1}}+\cdots+{{n-s}\choose{k-1}}\leq
s{{n-1}\choose{k-1}}=\frac{sk}{n}{{n}\choose{k}}.\hskip 36.98857pt$

Since a family $\mathcal{F}$ has matching number $\nu(\mathcal{F})\leq s$ is a
hereditary property, by applying Theorems 1.14 and 4.6 with

$f^{*}(n,k,\frac{sk}{n})=\frac{sk}{n}{{n}\choose{k}},$

we obtain

$|\mathcal{V}|\leq
f^{*}_{q}(n,k,\frac{sk}{n})=\frac{sk}{n}\genfrac{[}{]}{0.0pt}{}{n}{k}.$ (11)

Since
$\lim_{n\longrightarrow\infty}\frac{sk}{n}\frac{q^{n}-1}{(q^{k}-1)q^{n-k}}=0$,
it follows from (9) and (11) that for $n$ sufficiently large,

$|\mathcal{V}|\leq\frac{sk}{n}\genfrac{[}{]}{0.0pt}{}{n}{k}=\frac{sk}{n}\cdot\frac{q^{n}-1}{q^{k}-1}\genfrac{[}{]}{0.0pt}{}{n-1}{k-1}\hskip
176.40707pt$
$=\frac{sk}{n}\frac{q^{n}-1}{(q^{k}-1)q^{n-k}}q^{n-k}\genfrac{[}{]}{0.0pt}{}{n-1}{k-1}<q^{n-k}\genfrac{[}{]}{0.0pt}{}{n-1}{k-1}=\genfrac{[}{]}{0.0pt}{}{n}{k}-\genfrac{[}{]}{0.0pt}{}{n-1}{k}.$

$\hfill\Box$

Recently, Ihringer [20] provided the following conjecture about the largest
$s$-EM-family of $k$-spaces.

###### Conjecture 4.8

(Ihringer [20]) For $n\geq 2k$, if $Y$ is a largest $s$-EM-family of
$k$-spaces of $\mathbb{F}_{q}^{n}$, then $Y$ is the union of $s$ intersecting
families or its complement.

Ihringer [20] proved that this conjecture holds if
$16s\leq\min\\{q^{\frac{n-k-l+2}{3}},q^{\frac{n-k-r}{3}},q^{\frac{n}{2}-k+1}\\}$.

The next conjecture given by Aharoni and Howard [1] and Huang, Loh and Sudakov
[19], independently, extends the Erdős Matching Conjecture to a rainbow
(multipartite) setting.

###### Conjecture 4.9

(Rainbow Erdős Matching Conjecture [1] and [19]). Let $n\geq(s+1)k$ and let
$\mathcal{F}_{1},\mathcal{F}_{2},\dots,\mathcal{F}_{s+1}\subseteq{{[n]}\choose{k}}$.
If there are no pairwise disjoint $F_{1},\dots,F_{s+1}$, where
$F_{i}\in\mathcal{F}_{i}$ for every $i=1,\dots,s+1$, then

$\min_{i\in[s+1]}|\mathcal{F}_{i}|\leq\max\bigg{\\{}{{(s+1)k-1}\choose{k}},\mbox{
}{{n}\choose{k}}-{{n-s}\choose{k}}\bigg{\\}}.$

We call this conjecture the REMC for short. Huang, Loh and Sudakov [19] proved
that the REMC is true for $n\geq 3k^{2}s$, and recently, Lu, Wang and Yu [25]
verified the REMC for $n>2sk$ and $s$ sufficiently large, and Keller and
Lifshitz [22] proved the REMC for $n>Ck$, where $C$ depends only on $s$.

###### Theorem 4.10

(Huang, Loh and Sudakov [19], Keller and Lifshitz [22], and Lu, Wang and Yu
[25]). Assume that $\mathcal{F}_{1},\mathcal{F}_{2},$
$\dots,\mathcal{F}_{s+1}\subseteq{{[n]}\choose{k}}$. If there are no pairwise
disjoint $F_{1},\dots,F_{s+1}$, where $F_{i}\in\mathcal{F}_{i}$ for every
$i=1,\dots,s+1$ and $n>Ck$ with $C$ depending on $s$, then

$\min_{i\in[s+1]}|\mathcal{F}_{i}|\leq\max\bigg{\\{}{{(s+1)k-1}\choose{k}},\mbox{
}{{n}\choose{k}}-{{n-s}\choose{k}}\bigg{\\}}.$

Note that forbidding rainbow matching of size $s+1$ is a hereditary property.
By applying Theorems 1.14 and 4.10 and using the identities (8) and (9), we
obtain the next asymptotic $q$-analogue of Theorem 4.10 similar to the proof
of Theorem 4.7.

###### Theorem 4.11

Assume that $\mathcal{V}_{1},\mathcal{V}_{2},\dots,\mathcal{V}_{s+1}$ are
families of $k$-dimensional subspaces of $\mathbb{F}_{q}^{n}$. If there are no
pairwise disjoint $V_{1},\dots,V_{s+1}$, where $V_{i}\in\mathcal{V}_{i}$ for
every $i=1,\dots,s+1$, then

$\min_{i\in[s+1]}|\mathcal{V}_{i}|\leq\genfrac{[}{]}{0.0pt}{}{n}{k}-\genfrac{[}{]}{0.0pt}{}{n-1}{k}$

for $n$ sufficiently large.

##  5 Concluding Remarks

In section 2, we derived an upper bound for an intersecting $k$-Sperner family
of subspaces of $\mathbb{F}_{q}^{n}$ which gives a $q$-analogue of Erdős’
$k$-Sperner Theorem. In section 3, we provided a general connection between
upper bounds for the sizes of $\mathcal{L}$-intersecting or $T$-configuration
forbidden families of subsets of $[n]$ and upper bounds for the sizes of
$\mathcal{L}$-intersecting or $T$-configuration forbidden families of
subspaces of $\mathbb{F}_{q}^{n}$. As applications of Theorems 1.13 and 1.14,
we provided some results on families of subspaces of $\mathbb{F}_{q}^{n}$
which follow asymptotically from the corresponding theorems on families of
subsets of $[n]$, including generalizations of existing results about the
well-known Erdős matching conjecture and Erdős-Chvátal simplex conjecture.

We end this section with the following open problems which are $q$-analogues
of Erdős-Chvátal Simplex Conjecture and Erdős Matching Conjecture.

###### Conjecture 5.1

($q$-Analogue Erdős-Chvátal Simplex Conjecture). Let $k\geq d+1\geq 3$ and
$n\geq\frac{k(d+1)}{d}$. Suppose that $\mathcal{V}$ is a family of
$k$-dimensional subspaces of $\mathbb{F}_{q}^{n}$ with no $d$-simplex. Then

$|\mathcal{V}|\leq
q^{n-k}\genfrac{[}{]}{0.0pt}{}{n-1}{k-1}=\genfrac{[}{]}{0.0pt}{}{n}{k}-\genfrac{[}{]}{0.0pt}{}{n-1}{k}.$

###### Conjecture 5.2

($q$-Analogue Erdős Matching Conjecture). Suppose that $\mathcal{V}$ is a
family of $k$-dimensional subspaces of $\mathbb{F}_{q}^{n}$ with $n\geq(s+1)k$
and $\nu_{q}(\mathcal{V})\leq s$. Then

$|\mathcal{V}|\leq\min\bigg{\\{}\genfrac{[}{]}{0.0pt}{}{n}{k}-\genfrac{[}{]}{0.0pt}{}{n-1}{k},\mbox{
}s\genfrac{[}{]}{0.0pt}{}{n-1}{k-1}\bigg{\\}}$

Theorems 4.3 and 4.7 show that the conjectures above are true when $n$ is
sufficiently large.

## Declaration of Competing Interest

The authors declare that they have no conflicts of interest to this work.

## Acknowledgement

The research is supported by the National Natural Science Foundation of China
(71973103, 11861019, 12271527, 12071484). Guizhou Talent Development Project
in Science and Technology (KY[2018]046), Natural Science Foundation of Guizhou
([2019]1047, [2020]1Z001, [2021]5609).

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