1 Introduction

One of the first results of extremal finite set theory is Sperner’s theorem [13] that states that if for any pair \(F,F'\) of distinct sets in a set systems \({\mathcal F}\subseteq 2^{[n]}\) we have \(\min \{|F\setminus F'|,|F'\setminus F|\}\ge 1\), then \(|{\mathcal F}| \le \left( {\begin{array}{c}n\\ \lfloor n/2\rfloor |\end{array}}\right) \) 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 \(1\le \min \{|F\setminus G||,|G\setminus F|\}\le k\). 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, 6]). They obtained some structural results from which they deduced the following extremal theorem. For a set \(F\subseteq [n]=\{1,2,\ldots ,n\}\), 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 et al. [5]) If the set system \(\{\emptyset \}\ne \{F_1,F_2\ldots ,F_m\}\subseteq 2^{[n]}\) is 1-close Sperner, then the characteristic vectors \(v_{F_1},v_{F_2},\ldots ,v_{F_m}\) are linearly independent over \(\mathbb {R}\). In particular, \(m\le n\).

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 \(\min \{|F\setminus G|,|G\setminus F|\}\in L\). Note that being \(\{k\}\)-close Sperner is equivalent to being k-close Sperner, and we will use the latter notation. Our first result is the following.

Theorem 1.2

If the set system \(\{F_1,F_2\ldots ,F_m\}\subseteq 2^{[n]}\) is L -close Sperner for some \(L\subseteq [n]\) , then we have \(m\le \sum _{h=0}^{|L|}\left( {\begin{array}{c}n\\ h\end{array}}\right) \) . Furthermore, if \(|L|=1\) , then \(m\le n\) holds.

Note that if |L| is fixed and n tends to infinity, then the bound is asymptotically sharp as shown by \(L=\{1,2,\ldots ,k\}\) (i.e. the k-close Sperner property) and the set system \(\left( {\begin{array}{c}[n]\\ k\end{array}}\right) =\{F\subseteq [n]:|F|=k\}\). Observe also that the inequality \(m\le n\) is sharp for \(L=\{1\}\) as shown by the family of singletons, but there exist many other 1-close Sperner systems with n sets: the family of all co-singletons (complements of singletons) is 1-close Sperner. Also, if \(A_1\cup A_2\) is a partition of [n], then the family \(\{\{a\}:a\in A_1\} \cup \{A_2\setminus \{a\}: a\in A_2\}\) is 1-close Sperner and of size n. 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 \(m\le n\) 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 \(F,F'\) of distinct sets in \({\mathcal F}\) we have \(|F\cap F'|\in L\). Frankl and Wilson [10] proved the same upper bound \(\sum _{h=0}^{|L|}\left( {\begin{array}{c}n\\ h\end{array}}\right) \) 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 [1, 2]) turned out to be very effective in obtaining L-intersection theorems. In the proof of the furthermore 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 \(p_1(x),p_2(x),\ldots ,p_m(x)\in \mathbb {F}^n[x]\) be polynomials and \(v_1,v_2,\ldots , v_m\in \mathbb {F}^n\) be vectors such that \(p_i(v_i)\ne 0\) and \(p_i(v_j)=0\) holds for all \(1\le j<i\le m\). Then the polynomials are linearly independent.

Proof

Suppose that \(\sum _{i=1}^mc_ip_i(x)=0\). As \(p_i(v_1)=0\) for all \(1<i\) we obtain \(c_1p_1(v_1)=0\) 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 \(c_jp_j(v_j)= 0\) and therefore \(c_j=0\). \(\square \)

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 distances. 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 \(\lfloor n/2\rfloor =sd(A,C)\not \le sd(A,B)+sd(B,C)=1+1\)

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 furthermore part of Theorem 1.2 does not remain valid in this case. In a different context Frankl, Füredi, and Pach considered the case \(L=\{0,1,\ldots ,t\}\). They considered the following construction: let \(\emptyset =C_0\subsetneq C_1 \subsetneq C_2 \subsetneq \ldots \subsetneq C_n=[n]\) be a maximal chain and let

$$\begin{aligned} {\mathcal F}_{n,t}=\left\{ F: C_{|F|-t}\subset F\right\} \cup \left\{ F: |F|\le t ~\text {or}\ |F|\ge n-t\right\} . \end{aligned}$$

The size of \({\mathcal F}_{n,t}\) is \(\left( {\begin{array}{c}n\\ t+1\end{array}}\right) -\left( {\begin{array}{c}2t+1\\ t+1\end{array}}\right) +2\sum _{i=0}^t\left( {\begin{array}{c}n\\ i\end{array}}\right) \) and clearly \({\mathcal F}_{n,t}\) is \(\{0,1,\ldots ,t\}\)-sd. This gives the lower bounds in the following results.

Theorem 1.4

(Frankl et al. [9]) If \(n\ge 3\), we have \(ex_{sd}(n,\{0,1\})=\left( {\begin{array}{c}n\\ 2\end{array}}\right) +2n-1\).

Theorem 1.5

(Frankl et al. [9]) For any n, t with \(n\ge 2(t+2)\), we have

\(\left( {\begin{array}{c}n\\ t+1\end{array}}\right) -\left( {\begin{array}{c}2t+1\\ t+1\end{array}}\right) +2\sum _{i=0}^t\left( {\begin{array}{c}n\\ i\end{array}}\right) \le ex_{sd}(n,\{0,1,\ldots ,t\})<\left( {\begin{array}{c}n\\ t+1\end{array}}\right) +5(t+1)^2\left( {\begin{array}{c}n\\ t\end{array}}\right) \).

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 \(\sum _{i=1}^nu_iv_i\) 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 \(|F|\le |G|\) holds.

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

$$\begin{aligned} p'_{F,L}(x)=\prod _{h\in L}\left( |F|-v_F\cdot x-h\right) . \end{aligned}$$

We obtain \(p_{F,L}(x)\) from \(p'_{F,L}(x)\) by replacing every \(x_i^t\) term by \(x_i\) for every \(t\ge 2\) and \(i=1,2,\ldots ,n\). As \(0=0^t\) and \(1=1^t\) for any \(t\ge 2\), we have \(p_{F,L}(v_G)=p'_{F,L}(v_G)=\prod _{h\in L}(|F\setminus G|-h)\). Finally, observe that the polynomials \(p_{F,L}(x)\) all belong to the subspace \(M_{|L |}\) of \(\mathbb {R}^n[x]\) spanned by \(\{x_{i_1}x_{i_2}\ldots x_{i_l}:0\le l\le |L|,i_1<i_2<\ldots <i_l\}\), where \(l=0\) refers to the constant 1 polynomial \(\mathbf {1}\). Note that \(\dim (M_{|L|})=\sum _{i=0}^{|L|}\left( {\begin{array}{c}n\\ i\end{array}}\right) \).

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

Theorem 2.1

If the set system \(\{F_1,F_2\ldots ,F_m\}\subseteq 2^{[n]}\) is L-close Sperner, then the polynomials \(p_{F_1,L}(x),p_{F_2,L}(x),\ldots ,p_{F_m,L}(x)\) are linearly independent in \(\mathbb {R}^n[x]\). In particular, \(m\le \sum _{h=0}^{|L|}\left( {\begin{array}{c}n\\ h\end{array}}\right) \). Moreover, if \(|L|=1\) and \(\{F_1,F_2\ldots ,F_m\}\ne \{\emptyset \}\), then the polynomials \(p_{F_1,L}(x),p_{F_2,L}(x),\ldots ,p_{F_m,L}(x)\) are linearly independent in \(\mathbb {R}^n[x]\) even together with \(\mathbf {1}\). In particular, \(m\le n\).

Proof

We claim that if \(F_1,F_2,\ldots ,F_m\) are listed in a non-increasing order according to the sizes of the sets, then the polynomials \(p_{F_1,L}(x),p_{F_2,L}(x),\ldots ,p_{F_m,L}(x)\) and the characteristic vectors \(v_{F_1},v_{F_2},\ldots , v_{F_m}\) 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 \(p_{F,L}(v_{F})\ne 0\) holds for any \(F\subseteq [n]\), while if \(|F_j|\le |F_i|\), 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 \(L=\{s\}\), \({\mathcal F}=\{F_1,F_2,\ldots ,F_m\}\) and let us suppose towards a contradiction that \(\mathbf {1}=\sum _{i=1}^mc_{F_i}p_{F_i,L}(x)\) 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

$$\begin{aligned} 1=c_Fp_{F,L}(v_F)+\sum _{\begin{array}{c} F'\in {\mathcal F}\\ |F'|>|F| \end{array}}c_{F'}p_{F',L}(v_F), \end{aligned}$$
(1)

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 \(i<j\le j^*\). 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 \(p_{F',L}(v_F)=|F'|-|F|+s-s=|F'|-|F|\) provided \(|F'|\ge |F|\), we obtain

$$\begin{aligned} 1=c_Fp_{F,L}(v_F)+\sum _{\begin{array}{c} F'\in {\mathcal F}\\ |F'|>|F| \end{array}}c_{F'}p_{F',L}(v_F)=-sc_F+\sum _{j>i}c_jm_j(j-i). \end{aligned}$$
(2)

This shows that \(c_F\) does not depend on F but 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 \(\mathbf {1}=\sum _{i=1}^mc_{F_i}p_{F_i,L}(x)\), 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 \(\{F_1,F_2\ldots ,F_m\}\ne \{\emptyset \}\) is used.) \(\square \)

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 \({\mathcal F}'\subseteq 2^{[n]}\) with \(|{\mathcal F}|=|{\mathcal F}'|\) and \({\mathcal F}'\) containing sets of size between k and \(n-k\). Is it true that for such set systems we have \(\langle p_{F,[k]}: ~F\in {\mathcal F}'\rangle \cap M_{k-1}=\{\mathbf {0}\}\)? This would imply \(ex_{sd}(n,[k])=\left( {\begin{array}{c}n\\ k\end{array}}\right) \).

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 \(\left( {\begin{array}{c}n\\ 2\end{array}}\right) +2n-1\) by induction on n. Since \(\left( {\begin{array}{c}3\\ 2\end{array}}\right) +2\cdot 3-1=2^3\), the statement is trivially true for \(n=3\). Now assume that \(n\ge 4\) 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 \(|{\mathcal F}_i|\ge 3\), 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 \(|C_i\backslash C_j|\le 1\).

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.) \(\square \)

Let \(C_{p_1}, C_{p_2}, \ldots C_{p_t}\) (\(p_1<\ldots <p_t\)) denote the representative sets of the nonempty levels among \({\mathcal F}_1, {\mathcal F}_2, \ldots {\mathcal F}_{n-1}\). Since

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

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

$$\begin{aligned} {\mathcal G}=\left\{ F\backslash \{n\}~|~F\in {\mathcal F}\right\} ,~~~{\mathcal H}=\left\{ H\in 2^{[n-1]}~|~H,H\cup \{n\}\in {\mathcal F}\right\} . \end{aligned}$$

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, \(1\ge |(A\cup \{n\})\backslash B|=|A\backslash B|+1\). Therefore we have \(|A\backslash B|=0\) or equivalently \(A\subset B\). \(\square \)

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. \(\square \)

Claim 2.5

\(|{\mathcal H}|\le n+1\).

Proof

Let \({\mathcal H}_i=\{H\in {\mathcal H}:|H|=i\}\) for all \(i=0,1,\ldots , n-1\). If there is no i such that \(|{\mathcal H}_i|>1\), then \(|{\mathcal H}|\le n\). 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 \({\mathcal H}_{t+1}\), \({\mathcal H}_{t+2}, \ldots ,\) \({\mathcal H}_{t+k-2}\) 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 \({\mathcal H}_{t-k+2}\), \({\mathcal H}_{t-k+3}, \ldots ,\) \({\mathcal H}_{t-1}\) are all empty. In either case we get that \(|{\mathcal H}|\le k + (k-2)\cdot 0+ (n-k+1)\cdot 1=n+1\). \(\square \)

Now we can complete the proof of the theorem:

$$\begin{aligned} |{\mathcal F}|=|{\mathcal G}|+|{\mathcal H}|\le \left( {\begin{array}{c}n-1\\ 2\end{array}}\right) +2(n-1)-1+n+1=\left( {\begin{array}{c}n\\ 2\end{array}}\right) +2n-1. \end{aligned}$$

\(\square \)

Let us make two final remarks.

  • Observe that for the set \(L_\ell =\{\ell +1,\ell +2,\ldots ,n\}\) 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 Sect. 4 of [12].

  • A natural generalization arises in \(Q^n=\{0,1,\ldots ,q-1\}^n\). One can partially order \(Q^n\) by \(a\le b\) if and only if \(a_i\le b_i\) for all \(i=1,2,\ldots ,n\). We say that \(A\subseteq \{0,1,\ldots ,q-1\}^n\) 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 \(2\le i\le n\), \(1\le h\le q-1\) let \((v_{i,h})_i=h\), \((v_{i,h})_1=q-h+1\) and \((v_{i,h})_j=0\) if \(j\ne i,1\). Then it is easy to verify that \(\{v_{i,h}:2\le i\le n, 1\le h\le q-1\}\) 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 \(a=\{a_1, a_2,\ldots a_n\}\in Q^n\) let us define the set \(F_a\subseteq [(q-1)n]\) as follows.

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

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

    $$\begin{aligned} |A|=|A'|\le \sum _{h=0}^{q-1}\left( {\begin{array}{c}(q-1)n\\ h\end{array}}\right) =O_q\left( n^{q-1}\right) . \end{aligned}$$

    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 \(C_qn\).