On $L$-close Sperner systems

TL;DR

This paper studies the maximum size of special set systems called $L$-close Sperner systems, generalizing known results and providing new bounds for various choices of $L$, including cases with zero skew distance.

Contribution

It rederives extremal bounds for $L$-close Sperner systems and extends these results to broader classes of $L$, including cases with zero skew distance.

Findings

Reproved bounds for $L= ext{\{1\}}$

Generalized bounds for arbitrary $L$ with $|L|=1$

Reproved a theorem for $L=\{0,1\}$

Abstract

For a set $L$ of positive integers, a set system $\mathcal{F} \subseteq 2^{[n]}$ is said to be $L$-close Sperner, if for any pair $F,G$ of distinct sets in $\mathcal{F}$ the skew distance $sd(F,G)=\min\{|F\setminus G|,|G\setminus F|\}$ belongs to $L$. We reprove an extremal result of Boros, Gurvich, and Milani\v c on the maximum size of $L$-close Sperner set systems for $L=\{1\}$ and generalize to $|L|=1$ and obtain slightly weaker bounds for arbitrary $L$. We also consider the problem when $L$ might include 0 and reprove a theorem of Frankl, F\"uredi, and Pach on the size of largest set systems with all skew distances belonging to $L=\{0,1\}$.

Equations

$${\mathcal{F}}_{n,t}=\{F:C_{|F|-t}\subset F\}\cup \{F:|F|\le t\text{or}|F|\ge n-t\}.$$

$$p_{F,L}^{\prime }(x)=\prod _{h\in L}(|F|-v_F\cdot x-h).$$

$$1=c_F{p}_{F,L}(v_F)+\sum _{\begin{array}{c}{F}^{\prime }\in \mathcal{F}\\ |F^{\prime }|>|F|\end{array}}{c}_{F^{\prime }}{p}_{F^{\prime },L}(v_F),$$

$$1=c_F{p}_{F,L}(v_F)+\sum _{\begin{array}{c}{F}^{\prime }\in \mathcal{F}\\ |F^{\prime }|>|F|\end{array}}{c}_{F^{\prime }}{p}_{F^{\prime },L}(v_F)=-s{c}_F+\sum _{j>i}{c}_j{m}_j(j-i).$$

$$\left|\bigcup _{i=1}^{t-1}{C}_{p_i}\backslash C_{p_{i+1}}\right|\le \sum _{i=1}^{t-1}|C_{p_i}\backslash C_{p_{i+1}}|\le t-1\le n-2,$$

$$\mathcal{G}=\{F\backslash \{n\}|F\in \mathcal{F}\},\mathcal{H}=\{H\in 2^{[n-1]}|H,H\cup \{n\}\in \mathcal{F}\}.$$

$$|\mathcal{F}|=|\mathcal{G}|+|\mathcal{H}|\le \left(\genfrac{}{}{0ex}{}{n-1}{2}\right)+2(n-1)-1+n+1=\left(\genfrac{}{}{0ex}{}{n}{2}\right)+2n-1.$$

$$F_a:=\bigcup _{i=1}^n\bigcup _{j=1}^{a_i}\{(q-1)(i-1)+j\}$$

$$|A|=|A^{\prime }|\le \sum _{h=0}^{q-1}\left(\genfrac{}{}{0ex}{}{(q-1)n}{h}\right)=O_q(n^{q-1}).$$

Full text

1 Introduction

One of the first results of extremal finite set theory is Sperner’s theorem [13] that states that if for any pair $F,F^{\prime}$ of distinct sets in a set systems ${\mathcal{F}}\subseteq 2^{[n]}$ we have $\min\{|F\setminus F^{\prime}|,|F^{\prime}\setminus F|\}\geq 1$, then $|{\mathcal{F}}|\leq\binom{n}{\lfloor n/2\rfloor|}$ holds. Set systems with this property are called antichains or Sperner systems. This theorem has lots of generalizations and applications in different areas of mathematics (see the book [7] and Chapter 3 of [11]). Recently, Boros, Gurvich, and Milanič introduced the following notion: given a positive integer $k$, we say that a set system ${\mathcal{F}}$ is $k$-close Sperner if every pair $F,G\in{\mathcal{F}}$ of distinct sets satisfies $1\leq\min\{|F\setminus G||,|G\setminus F|\}\leq k$. In particular, ${\mathcal{F}}$ is 1-close Sperner if every pair $F,G\in{\mathcal{F}}$ of distinct sets satisfies $\min\{|F\setminus G||,|G\setminus F|\}=1$. (The authors used the unfortunate $k$-Sperner term which, throughout the literature, refers to set systems that are union of $k$ many antichains. That is why we decided to use instead the terminology $k$-close Sperner systems.) Boros, Gurvich, and Milanič’s motivation to study these set systems comes from computer science: they wanted to compare them to other classes of Sperner systems (see also [4] and [6]). They obtained some structural results from which they deduced the following extremal theorem. For a set $F\subseteq[n]=\{1,2,\dots,n\}$, its characteristic vector $v_{F}$ is a 0-1 vector of length $n$ with $(v_{F})_{i}=1$ if and only if $i\in F$.

Theorem 1.1** (Boros, Gurvich, Milanič [5]).**

If the set system $\{\emptyset\}\neq\{F_{1},F_{2}\dots,F_{m}\}\subseteq 2^{[n]}$ is 1-close Sperner, then the characteristic vectors $v_{F_{1}},v_{F_{2}},\dots,v_{F_{m}}$ are linearly independent over $\mathbb{R}$. In particular, $m\leq n$.

In this short note, we reprove the extremal part of Theorem 1.1 via a different linear algebraic approach and generalize the result. For a subset $L$ of $[n]$, we say that a set system ${\mathcal{F}}$ is $L$-close Sperner if every pair $F,G\in{\mathcal{F}}$ satisfies $\min\{|F\setminus G|,|G\setminus F|\}\in L$. Our first result is the following.

Theorem 1.2**.**

If the set system $\{F_{1},F_{2}\dots,F_{m}\}\subseteq 2^{[n]}$ is $L$-close Sperner for some $L\subseteq[n]$, then we have $m\leq\sum_{h=0}^{|L|}\binom{n}{h}$. Furthermore, if $|L|=1$, then $m\leq n$ holds.

Note that if $|L|$ is fixed and $n$ tends to infinity, then the bound is asymptotically sharp as shown by $L=\{1,2,\dots,k\}$ (i.e. the $k$-close Sperner property) and the set system $\binom{[n]}{k}=\{F\subseteq[n]:|F|=k\}$. Observe also that the inequality $m\leq n$ is sharp for $L=\{1\}$ as shown by the family of singletons, but there exist many other $1$-close Sperner systems with $n$ sets. Furthermore, if $L=\{q\}$ for some prime power $q$ and $n=q^{2}+q+1$, then the lines of a projective plane of order $q$ form an $L$-close family of size $n$, so the bound $m\leq n$ is sharp in this case, too.

Apart from Sperner-type theorems, the other much studied area in extremal finite set theory are intersection properties (see e.g. Chapter 2 of [11]). For a set $L$ of integers, a set system ${\mathcal{F}}$ is said to be $L$-intersecting if for any pair $F,F^{\prime}$ of distinct sets in ${\mathcal{F}}$ we have $|F\cap F^{\prime}|\in L$. Frankl and Wilson [10] proved the same upper bound $\sum_{h=0}^{|L|}\binom{n}{h}$ on the size of $L$-intersecting set systems. Frankl and Wilson used higher incidence matrices to prove their result, but later the polynomial method (see [2] and [1]) turned out to be very effective in obtaining $L$-intersection theorems. In the proof of the moreover part of Theorem 1.2, an additional idea due to Blokhuis [3] will be used.

We will need the following well-known lemma, we include the proof for sake of completeness. For any field $\mathbb{F}$, we denote by $\mathbb{F}^{n}[x]$ the vector space over $\mathbb{F}$ of polynomials of $n$ variables with coefficients from $\mathbb{F}$.

Lemma 1.3**.**

Let $p_{1}(x),p_{2}(x),\dots,p_{m}(x)\in\mathbb{F}^{n}[x]$ be polynomials and $v_{1},v_{2},\dots,v_{m}\in\mathbb{F}^{n}$ be vectors such that $p_{i}(v_{i})\neq 0$ and $p_{i}(v_{j})=0$ holds for all $1\leq j<i\leq m$. Then the polynomials are linearly independent.

Proof.

Suppose that $\sum_{i=1}^{m}c_{i}p_{i}(x)=0$. As $p_{i}(v_{1})=0$ for all $1<i$ we obtain $c_{1}p_{1}(v_{1})=0$ and therefore $c_{1}=0$ holds. We proceed by induction on $j$. If $c_{h}=0$ holds for all $h<j$, then using this and $p_{i}(v_{j})=0$ for all $i>j$, we obtain $c_{j}p_{j}(v_{j})=0$ and therefore $c_{j}=0$. ∎

Results on $L$-intersecting families had some geometric consequences on point sets in $\mathbb{R}^{n}$ defining only a few distances, in particular on set systems ${\mathcal{F}}$ with only a few Hamming distance. The skew distance $sd(F,G):=\min\{|F\setminus G|,|G\setminus F|\}$ does not define a metric space on $2^{[n]}$ as $sd(F,G)=0$ holds if and only if $F\subseteq G$ or $G\subseteq F$ and one can easily find triples for which the triangle inequality fails: if $A$ is the set of even integers in $[n]$, $C$ is the set of odd integers in $[n]$, and $B=\{1,2\}$, then $\lfloor n/2\rfloor=sd(A,C)\not\leq sd(A,B)+sd(B,C)=1+1$

One can also investigate the case when $L$ includes 0. Then set systems with the required property are not necessarily Sperner, so we will say that ${\mathcal{F}}$ is $L-$skew distance (or $L$-sd for short) if $sd(A,B)\in L$ for all pairs of distinct sets $A,B\in{\mathcal{F}}$. We will write $ex_{sd}(n,L)$ to denote the largest size of an $L$-skew distance system ${\mathcal{F}}\subseteq 2^{[n]}$. Observe that $ex_{sd}(n,\{0\})$ asks for the maximum size of a chain in $2^{[n]}$ which is obviously $n+1$. This shows that the moreover part of Theorem 1.2 does not remain valid in this case. In a different context Frankl, Füredi, and Pach considered the case $L=\{0,1,\dots,t\}$. They considered the following construction: let $\emptyset=C_{0}\subsetneq C_{1}\subsetneq C_{2}\subsetneq\dots\subsetneq C_{n}=[n]$ be a maximal chain and let

[TABLE]

The size of ${\mathcal{F}}_{n,t}$ is $\binom{n}{t+1}-\binom{2t+1}{t+1}+2\sum_{i=0}^{t}\binom{n}{i}$ and clearly ${\mathcal{F}}_{n,t}$ is $\{0,1,\dots,t\}$-sd. This gives the lower bounds in the following results.

Theorem 1.4** (Frankl, Füredi, Pach, [9]).**

If $n\geq 3$, we have $ex_{sd}(n,\{0,1\})=\binom{n}{2}+2n-1$.

Theorem 1.5** (Frankl, Füredi, Pach, [9]).**

*For any $n,t$ with $n\geq 2(t+2)$, we have

$\binom{n}{t+1}-\binom{2t+1}{t+1}+2\sum_{i=0}^{t}\binom{n}{i}\leq ex_{sd}(n,\{0,1,\dots,t\})<\binom{n}{t+1}+5(t+1)^{2}\binom{n}{t}$.*

The authors of [9] conjectured that the lower bound is tight in Theorem 1.5 for large enough $n$. (There are larger constructions for small $n$.) We will give a simple, new proof of Theorem 1.4 that proceeds by induction.

2 Proof and remarks

We start by introducing some notation. For two vectors, $u,v$ of length $n$ we denote their scalar product $\sum_{i=1}^{n}u_{i}v_{i}$ by $u\cdot v$. We will often use the fact that for any pair $F,G$ of sets we have $v_{F}\cdot v_{G}=|F\cap G|$. We will also use that $\min\{|F\setminus G|,|G\setminus F|\}=|F\setminus G|$ if and only if $|F|\leq|G|$ holds.

For two sets $F,L\subseteq[n]$ we define the polynomial $p^{\prime}_{F,L}\in\mathbb{R}^{n}[x]$ as

[TABLE]

We obtain $p_{F,L}(x)$ from $p^{\prime}_{F,L}(x)$ by replacing every $x_{i}^{t}$ term by $x_{i}$ for every $t\geq 2$ and $i=1,2,\dots,n$. As $0=0^{t}$ and $1=1^{t}$ for any $t\geq 2$, we have $p_{F,L}(v_{G})=p^{\prime}_{F,L}(v_{G})=\prod_{h\in L}(|F\setminus G|-h)$. Finally, observe that the polynomials $p_{F,L}(x)$ all belong to the subspace $M_{|L|}$ of $\mathbb{R}^{n}[x]$ spanned by $\{x_{i_{1}}x_{i_{2}}\dots x_{i_{l}}:0\leq l\leq|L|,i_{1}<i_{2}<\dots<i_{l}\}$, where $l=0$ refers to the constant 1 polynomial $\mathbf{1}$. Note that $\dim(M_{|L|})=\sum_{i=0}^{|L|}\binom{n}{i}$.

Based on the above, Theorem 1.2 is an immediate consequence of the next result.

Theorem 2.1**.**

If the set system $\{F_{1},F_{2}\dots,F_{m}\}\subseteq 2^{[n]}$ is $L$-close Sperner, then the polynomials $p_{F_{1},L}(x),p_{F_{2},L}(x),\dots,p_{F_{m},L}(x)$ are linearly independent in $\mathbb{R}^{n}[x]$. In particular, $m\leq\sum_{h=0}^{|L|}\binom{n}{h}$. Moreover, if $|L|=1$ and $\{F_{1},F_{2}\dots,F_{m}\}\neq\{\emptyset\}$, then the polynomials $p_{F_{1},L}(x),p_{F_{2},L}(x),\dots,p_{F_{m},L}(x)$ are linearly independent in $\mathbb{R}^{n}[x]$ even together with $\mathbf{1}$. In particular, $m\leq n$.

Proof.

We claim that if $F_{1},F_{2},\dots,F_{m}$ are listed in a non-increasing order according to the sizes of the sets, then the polynomials $p_{F_{1},L}(x),p_{F_{2},L}(x),\dots,p_{F_{m},L}(x)$ and the characteristic vectors $v_{F_{1}},v_{F_{2}},\dots,v_{F_{m}}$ satisfy the conditions of Lemma 1.3. Indeed, for any $G\subseteq[n]$ we have $p_{F,k}(G)=\prod_{h\in L}(|F|-|F\cap G|-h)=\prod_{h\in L}(|F\setminus G|-h)$. Therefore $p_{F,L}(v_{F})\neq 0$ holds for any $F\subseteq[n]$, while if $|F_{j}|\leq|F_{i}|$, then the $L$-close Sperner property ensures $|F_{i}\setminus F_{j}|\in L$ and thus $p_{F_{j},L}(v_{F_{i}})=0$.

To prove the moreover part, let $L=\{s\}$, ${\mathcal{F}}=\{F_{1},F_{2},\dots,F_{m}\}$ and let us suppose towards a contradiction that $\mathbf{1}=\sum_{i=1}^{m}c_{F_{i}}p_{F_{i},L}(x)$ holds for some reals $c_{F_{i}}$. We claim that if $|F_{i}|=|F_{j}|$, then $c_{F_{i}}=c_{F_{j}}$ holds and all coefficients are negative. Observe that for any $F\in{\mathcal{F}}$ using the $L$-close Sperner property we have

[TABLE]

and $p_{F,L}(v_{F})=-s$ for all $F$. In particular, if $F$ is of maximum size in ${\mathcal{F}}$, then $c_{F}=-\frac{1}{s}$ holds. Let $m_{j}$ denote $|\{F\in{\mathcal{F}}:|F|=j\}|$ and $c_{j}$ denote the value of $c_{F}$ for all $F\in{\mathcal{F}}$ of size $j$ - once this is proved. By the above, if $j^{*}$ is the maximum size among sets in ${\mathcal{F}}$, then $c_{j^{*}}$ exists. Suppose that for some $i$ we have proved the existence of $c_{j}$ for all $j$ with $i<j\leq j^{*}$. If there is no set in ${\mathcal{F}}$ of size $i$, there is nothing to prove. If $|F|=i$, then using (1) and the fact $p_{F^{\prime},L}(v_{F})=|F^{\prime}|-|F|+s-s=|F^{\prime}|-|F|$ provided $|F^{\prime}|\geq|F|$, we obtain

[TABLE]

This shows that $c_{F}$ does not depend on $F$ only on $|F|$ as claimed. Moreover, as $s$, $m_{j}$, $j-i$ are all non-negative and, by induction, all $c_{j}$ are negative, then in order to satisfy (2), we must have that $c_{i}$ is negative as well. So we proved that all $c_{j}$’s are negative. But this contradicts $\mathbf{1}=\sum_{i=1}^{m}c_{F_{i}}p_{F_{i},L}(x)$, as on the right hand side all coefficients of the variables are positive, so they cannot cancel. (If there are variables. This is where the condition $\{F_{1},F_{2}\dots,F_{m}\}\neq\{\emptyset\}$ is used.) ∎

Using the original ”push-to-the-middle” argument of Sperner, it is not hard to prove that for any $k$-close Sperner system ${\mathcal{F}}\subseteq 2^{[n]}$, there exists another one ${\mathcal{F}}^{\prime}\subseteq 2^{[n]}$ with $|{\mathcal{F}}|=|{\mathcal{F}}^{\prime}|$ and ${\mathcal{F}}^{\prime}$ containing sets of size between $k$ and $n-k$. Is it true that for such set systems we have $\langle p_{F,[k]}:~{}F\in{\mathcal{F}}^{\prime}\rangle\cap M_{k-1}=\{\mathbf{0}\}$? This would imply $ex_{sd}(n,[k])=\binom{n}{k}$.

Let us now turn to the proof of Theorem 1.4.

Proof of Theorem 1.4.

The lower bound is given by the special case $t=1$ of the construction given above Theorem 1.4. It remains to prove the upper bound.

We will prove that a $\{0,1\}$-sd system ${\mathcal{F}}\subseteq 2^{[n]}$ is of size at most $\binom{n}{2}+2n-1$ by induction on $n$. Since $\binom{3}{2}+2\cdot 3-1=2^{3}$, the statement is trivially true for $n=3$. Now assume that $n\geq 4$ and we have already proved the statement for $n-1$.

Consider the uniform systems ${\mathcal{F}}_{i}=\{F\in{\mathcal{F}}:|F|=i\}$ that are 1-close Sperner. We will define a representative set $C_{i}$ for all nonempty levels. If $|{\mathcal{F}}_{i}|\geq 3$, it is an exercise for the reader (see Lemma 19 in [5]) to see that there exists a set $C_{i}$ either with $|C_{i}|=i-1$ and $C_{i}\subseteq\cap_{F\in{\mathcal{F}}_{i}}F$ or with $|C_{i}|=i+1$ and $\cup_{F\in{\mathcal{F}}_{i}}F\subseteq C_{i}$. In the former case we say that ${\mathcal{F}}_{i}$ is of type $\vee$, in the latter case we say that ${\mathcal{F}}_{i}$ is of type $\wedge$. If $|{\mathcal{F}}_{i}|=2$, then we select one of the two sets to be $C_{i}$. If $|{\mathcal{F}}_{i}|=1$, then $C_{i}$ is the only set in ${\mathcal{F}}_{i}$. Finally, if ${\mathcal{F}}_{i}=\emptyset$, then $C_{i}$ is undefined.

Claim 2.2**.**

If $i<j$ and $|{\mathcal{F}}_{i}|,|{\mathcal{F}}_{j}|>0$ then $|C_{i}\backslash C_{j}|\leq 1$.

Proof.

Assume that there are two different elements $a,b$ such that $a,b\in C_{i}$ but $a,b\not\in C_{j}$. It follows from the definition of the representative sets, that there are sets $F_{i}\in{\mathcal{F}}_{i}$ and $F_{j}\in{\mathcal{F}}_{j}$ such that $a,b\in F_{i}$ and $a,b\not\in F_{j}$. (This is trivial for levels with one or two sets. If there are 3 or more sets then at most two of them can be wrong.) ∎

Let $C_{p_{1}},C_{p_{2}},\dots C_{p_{t}}$ ($p_{1}<\dots<p_{t}$) denote the representative sets of the nonempty levels among ${\mathcal{F}}_{1},{\mathcal{F}}_{2},\dots\mathcal{F}_{n-1}$. Since

[TABLE]

there will be an element $x\in[n]$ such that $x\not\in C_{p_{i}}\backslash C_{p_{i+1}}$ for any $p_{i}$. This implies that there are no nonempty levels ${\mathcal{F}}_{i}$ and ${\mathcal{F}}_{j}$ such that $i<j$, $x\in C_{i}$ but $x\not\in C_{j}$. Rearranging the names of the elements, we may assume that $x=n$.

Now we define two families in $2^{[n-1]}$, let

[TABLE]

Note that $|{\mathcal{F}}|=|{\mathcal{G}}|+|{\mathcal{H}}|$. Since ${\mathcal{G}}$ is a $\{0,1\}$-sd system in $2^{[n-1]}$, we get an upper bound on its size by induction. We will examine ${\mathcal{H}}$ to bound its size as well.

Claim 2.3**.**

If $A,B\in{\mathcal{H}}$ and $|A|<|B|$ then $A\subset B$.

Proof.

By the definition of ${\mathcal{H}}$, we get that $A\cup\{n\}\in{\mathcal{F}}$ and $n\not\in B$. Since ${\mathcal{F}}$ is a $\{0,1\}$-sd system, $1\geq|(A\cup\{n\})\backslash B|=|A\backslash B|+1$. Therefore we have $|A\backslash B|=0$ or equivalently $A\subset B$. ∎

Claim 2.4**.**

There is at most one level in ${\mathcal{H}}$ with two or more sets in it.

Proof.

Assume that there are two sets of size $i$ and two sets of size $j$ ($i<j$) in ${\mathcal{H}}$. Then in ${\mathcal{F}}$ there are two sets of size $i+1$ containing $n$ and two sets of size $j$ that do not contain $n$. From the definition of the representative sets follows that $n\in C_{i+1}$ but $n\not\in C_{j}$. This is an outright contradiction if $i+1=j$. If $i+1<j$, it contradicts the special property of the element $n$ established earlier. ∎

Claim 2.5**.**

$|{\mathcal{H}}|\leq n+1$.

Proof.

Let ${\mathcal{H}}_{i}=\{H\in{\mathcal{H}}:|H|=i\}$ for all $i=0,1,\dots,n-1$. If there is no $i$ such that $|{\mathcal{H}}_{i}|>1$, then $|{\mathcal{H}}|\leq n$. Assume that $|{\mathcal{H}}_{t}|=k>1$. By Claim 2.4, this is the only level with more than one set. If the level ${\mathcal{H}}_{t}$ is of type $\vee$, then the union of its sets is of size $t+k-1$. Claim 2.3 implies that all sets $H\in{\mathcal{H}}$, $|H|>t$ must contain this union, therefore the levels ${\mathcal{H}}_{t+1}$, ${\mathcal{H}}_{t+2},\dots,$ ${\mathcal{H}}_{t+k-2}$ are all empty. If ${\mathcal{H}}_{t}$ is of type $\wedge$, then the intersection of its sets is of size $t-k+1$. Claim 2.3 implies that all sets $H\in{\mathcal{H}}$, $|H|<t$ must be subsets of this intersection, therefore the levels ${\mathcal{H}}_{t-k+2}$, ${\mathcal{H}}_{t-k+3},\dots,$ ${\mathcal{H}}_{t-1}$ are all empty. In either case we get that $|{\mathcal{H}}|\leq k+(k-2)\cdot 0+(n-k+1)\cdot 1=n+1$. ∎

Now we can complete the proof of the theorem:

[TABLE]

∎

Let us make two final remarks.

• •

Observe that for the set $L_{\ell}=\{\ell+1,\ell+2,\dots,n\}$ a system ${\mathcal{F}}\subseteq 2^{[n]}$ is $L_{\ell}$-close Sperner if and only if for every $\ell$-subset $Y$ of $[n]$, the trace ${\mathcal{F}}_{[n]\setminus Y}=\{F\setminus Y:F\in{\mathcal{F}}\}$ is Sperner. Set systems with this property are called $(n-\ell)$-trace Sperner and results on the maximum size of such systems can be found in Section 4 of [12].

• •

A natural generalization arises in $Q^{n}=\{0,1,\dots,q-1\}^{n}$. One can partially order $Q^{n}$ by $a\leq b$ if and only if $a_{i}\leq b_{i}$ for all $i=1,2,\dots,n$. We say that $A\subseteq\{0,1,\dots,q-1\}^{n}$ is $L$-close Sperner for some subset $L\subseteq[n]$ if for any distinct $a,b\in A$ we have $sd(a,b):=\min\{|\{i:a_{i}<b_{i}\}|,|\{i:a_{i}>b_{i}\}|\}\in L$. One can ask for the largest number of points in an $L$-close Sperner set $A\subseteq Q^{n}$. Here is a construction for $\{1\}$-close Sperner set: for $2\leq i\leq n$, $1\leq h\leq q-1$ let $(v_{i,h})_{i}=h$, $(v_{i,h})_{1}=q-h+1$ and $(v_{i,h})_{j}=0$ if $j\neq i$. Then it is easy to verify that $\{v_{i,h}:2\leq i\leq n,1\leq h\leq q-1\}$ is $\{1\}$-close Sperner of size $(q-1)(n-1)$.

An easy upper bound on the most number of points in $Q^{n}$ that form an $\{1\}$-close Sperner system is $O_{q}(n^{q-1})$. To see this, for any $a=\{a_{1},a_{2},\dots a_{n}\}\in Q^{n}$ let us define the set $F_{a}\subseteq[(q-1)n]$ as follows.

[TABLE]

If $A\subseteq Q^{n}$ is $\{1\}$-close Sperner, then $A^{\prime}=\{F_{a}~{}|~{}a\in A\}\subset 2^{[(q-1)n]}$ will be $\{1,2,\dots q-1\}$-close Sperner. Theorem 1.2 implies

[TABLE]

We conjecture that for any $q$ there exists a constant $C_{q}$ such that the maximum number of points in $Q^{n}$ that form a $\{1\}$-close Sperner system is at most $C_{q}n$.

Acknowledgement

The research of Nagy was supported by the National Research, Development and Innovation Office - NKFIH under the grants FK 132060 and K 132696.

The research of Patkós was supported partially by the grant of Russian Government N 075-15-2019-1926 and by the National Research, Development and Innovation Office - NKFIH under the grants FK 132060 and SNN 129364.