Improved bounds for cross-Sperner systems

A collection of families $(\mathcal{F}_{1}, \mathcal{F}_{2} , \cdots , \mathcal{F}_{k}) \in \mathcal{P}([n])^k$ is cross-Sperner if there is no pair $i \not= j$ for which some $F_i \in \mathcal{F}_i$ is comparable to some $F_j \in \mathcal{F}_j$. Two natural measures of the `size' of such a family are the sum $\sum_{i = 1}^k |\mathcal{F}_i|$ and the product $\prod_{i = 1}^k |\mathcal{F}_i|$. We prove new upper and lower bounds on both of these measures for general $n$ and $k \ge 2$ which improve considerably on the previous best bounds. In particular, we construct a rich family of counterexamples to a conjecture of Gerbner, Lemons, Palmer, Patk\'{o}s, and Sz\'{e}csi from 2011.


Introduction
A family F ⊆ P ([n]) is an antichain (also known as a Sperner family) if for all distinct F, G ∈ F , neither F ⊆ G nor G ⊆ F (i.e.F and G are incomparable).One of the principal results in extremal combinatorics is Sperner's theorem [22], which states that the largest size of an antichain in P([n]) is n ⌊n/2⌋ .This can be seen to be tight by taking a 'middle layer', that is F = [n]  ⌊n/2⌋ or F = [n]  ⌈n/2⌉ .It is natural to consider a generalisation of Sperner's theorem to multiple families of sets.For k ≥ 2, say that a collection of non-empty families (F 1 , F 2 , • • • , F k ) ∈ P([n]) k , is cross-Sperner if for all i = j, the sets F i and F j are incomparable for any F i ∈ F i and F j ∈ F j .(We may also write that ( The study of such objects goes back to the 1970s when Seymour [21] deduced from a result of Kleitman [14] that a cross-Sperner pair (F , G) in P([n]) satisfies hence resolving a related conjecture of Hilton (see [3]).Equality is obtained in Seymour's bound precisely when the minimal sets of F are pairwise disjoint from the minimal sets intersecting each set of G.A broad spectrum of research concerning discrete objects with 'Sperner-like' properties have since emerged (see, for example, [1,2,4,5,7,10,11,12,13,19,23]).Many related results concern families satisfying both Sperner-type properties, and additional properties such as conditions on intersections (see, for example [6,15,17,18,20]).
).There are several natural measures of the 'size' of such a family.These include the sum The general study of these quantities was initiated by Gerbner, Lemons, Palmer, Patkós, and Szécsi [8], who essentially proved best possible bounds on cross-Sperner pairs of families.
Concerning the product, they gave a direct proof that a cross-Sperner pair (F , G) To see that this bound is tight, consider 2) can also be obtained as a direct consequence of (1.1) via the AM-GM inequality 1 .
First, let us focus on product bounds for k ≥ 3. It is convenient to define In [8], it was observed that (1.1) can be used to obtain the upper bound π(n, k) ≤ 2 k(n−2) .For k > 4, an improved bound of π(n, k) ≤ 2 n k k can be obtained by a simple application of the AM-GM inequality. 2 Gerbner, Lemons Palmer, Patkós, and Szécsi [8] conjectured that π(n, k) ≤ 2 k(n−ℓ * ) , where ℓ * = ℓ * (k) is the least positive integer such that ℓ * ⌊ℓ * /2⌋ ≥ k.They described a construction which provides a matching lower bound to their conjecture.Let A 1 , . . ., A k be an antichain in P([l]) and let (F 1 , . . ., F k ) ∈ P([n]) be defined by Our first theorem strongly disproves this conjecture.
2 Similarly to above, we have Theorem 1.3.Let n and k ≥ 2 be integers.For n sufficiently large, A crude application of Stirling's approximation yields that ℓ * (k) = ω(log k).So in particular, there is a function g(k) tending to infinity with k such that 2 k(n−ℓ * ) = O 2 kn (k • g(k)) −k .Therefore our lower bound is exponentially larger than the conjectured 2 k(n−ℓ * ) .
We also improve the previous best known upper bound by a factor of 2 k .
Theorem 1.4.Let n and k ≥ 2 be integers.Then Regarding bounds on the sum, in [8] it is shown that for n sufficiently large, a cross-Sperner pair in P([n]) satisfies This is tight, which can be seen by taking F = {1, 2, . . ., ⌊n/2⌋} and letting G be all subsets of [n] that are not comparable to F .Gerbner, Lemons Palmer, Patkós, and Szécsi [8] also asked about bounds for the sum for general k.Analogously to in the product case, define In our next theorem, we determine upper and lower bounds on σ(n, k).
Theorem 1.6.Let n, k be integers with n ≥ 2k.Then When k is a power of 2 and n − log 2 k is even, we can further improve the lower bound to 2 n − 2 √ 2 n k + 2(k − 1), which is extremely close to the upper bound.In order to prove Theorem 1.4 and the upper bound in Theorem 1.6, we exploit a connection between σ(n, k) and the comparability number of a set (given in Section 2).In doing so, we recover a simple proof of (1.5) (see Theorem 2.4) that holds for all n (recall the result of [8] holds for large n).
The article is structured as follows.We introduce the comparability number in Section 2 and provide a lower bound (Theorem 2.3) that will be used in the proofs of Theorems 1.4 and 1.6.In Section 3 we prove Theorems 1.3 and 1.4 bounding the product.In Section 4 we prove Theorem 1.6 bounding the sum.We conclude in section 5 with some discussion and open questions.

Minimizing Comparability
Given a family F ⊆ P([n]) define the comparability number of F to be When the setting is clear from context, we may write c(F ) for c(n, F ). Define As noted in [8], there is a direct relationship between σ(n, 2) and c(n, m).Observe that if (F , G) is cross-Sperner in P([n]), we have as any set incomparable to every member of F can be added to G. We will use analogous ideas in Section 4 to provide upper bounds on σ(n, k) for k ≥ 3.
Our goal in this section is to find a lower bound on c(n, m).We begin by showing that families that minimize comparability are 'convex'.
Proof of Lemma 2.1.Let Z be a set such that X ⊆ Z ⊆ Y , for some X, Y ∈ F .Observe that any set in P([n]) that is comparable to Z is either comparable to X or to Y .So c(F ∪ Z) = c(F ).Repeatedly applying this observation gives the result.Theorem 2.3 can now be deduced from the Harris-Kleitman inequality.Recall that a family Lemma 2.2 (Harris-Kleitman Inequality [14]).Let U ⊆ P([n]) be an upset and We will apply Lemma 2.2 to prove a lower bound on c(n, m).For convenience, for a family F ⊆ P([n]), define Proof.Let F ⊆ P([n]) be such that |F | = m and c(F ) = c(n, m).We may assume F is convex.If not, by Lemma 2.1 we may add sets to make it convex and then remove minimal or maximal elements to obtain Using the AM-GM inequality we get Since U F is an upset and D F is a downset, we apply Lemma 2.2 to get as required.
It is now a simple consequence of Theorem 2.3 to see that (1.5) holds for all n.We have the following two cases.
Case 1: Suppose m = 1.Since F only consists of one set, say F , we have as required.This completes the case m = 1.
Case 2: Now suppose m ≥ 2. By Theorem 2.3, By differentiation with respect to m we see that the expression on the right-hand side is decreasing in the range 2 ≤ m ≤ 2 n−2 .It is therefore maximized at m = 2, where we have Note that for all n ≥ 2, We conclude that |F | + |G| ≤ 2 n − 2 ⌊n/2⌋ − 2 ⌈n/2⌉ + 2, as desired.

Bounding π(n, k)
The goal of this section is to prove Theorems 1.3 and 1.4.

Lower Bound on π(n, k)
Theorem 1.3 follows directly from the following (slightly stronger) statement.
For each 1 ≤ i ≤ k, take X i to be an initial segment of colex in P(A i ) such that Refer to Example 3.4 for an example of this construction.
To see that ( ), consider S ∈ F i and T ∈ F j .We must show that S and T are incomparable.If S ⊆ T .Then S ∩ A j ⊆ T ∩ A j , so there is some Y ∈ Y j and X ∈ X j such that Y ⊆ X, a contradiction.Analogously, we see that T cannot be a subset of S.
Observe that and so To complete the proof of Lemma 3.1 it remains to optimise the sizes of the λ i .We have . We have For n > k log 2 k + k we have 2 −⌊n/k⌋ ≤ 2 −(n/k−1) < 1 k and so λ i is not zero.Therefore, with this choice of λ i we get 2 kn , as required.
Remark 3.3: Note that if k is a power of 2, in the proof of Lemma 3.1 we have λ i = 1 k for all 1 ≤ i ≤ k.Therefore in this case we can eliminate the − 1 2 ⌊n/k⌋ term.For clarity, we provide an example of the construction given in Lemma 3.1.
Example 3.4.Let n = 6 and k = 3. Partition [6] into Then let Then we construct our cross-Sperner system to be We now deduce Theorem 1.3 (restated below for convenience) from Lemma 3.1.
Theorem 1.3.Let n and k ≥ 2 be integers.For n sufficiently large, Proof.Take n sufficiently large so that

This is possible as 1 + 1
k−1 k−1 tends to e from below.Substituting this into Lemma 3.1, we see that

Upper Bound on π(n, k)
The goal of this subsection is to prove Theorem 1.4, restated below for convenience.
Theorem 1.4.Let n and k ≥ 2 be integers.Then We will use the following observation.
Lemma 3.5.Let 1 ≤ j < k and let (F 1 , F 2 , . . ., F k ) ⊆ P([n]) k be cross-Sperner.Then Proof.Suppose for contradiction that j i=1 F i , k i=j+1 F i is not cross-Sperner.Then there exists some X ∈ j i=1 F i and Y ∈ k i=j+1 F i such that X ⊆ Y or Y ⊆ X.Since X ∈ F i for some 1 ≤ i ≤ j, and Y ∈ F t for some j + 1 ≤ t ≤ k we deduce that (F 1 , F 2 , . . ., F k ) is not cross-Sperner, a contradiction.We now use Lemma 3.5, along with Theorem 2.3, to give an upper bound on π(n, k).
since each product is maximized when the families are of equal sizes.Thus, To find an upper bound on the left hand side of (3.6), we differentiate with respect to m to find the value of m that maximises the right hand side.
Setting this equal to zero yields m ∈ {0, 2 n , a 2 2 n k 2 }.A simple calculation shows that (3.6) is maximized when m = a 2 2 n k 2 .Thus , as required.
Note that for k even, the upper bound given by Theorem 1.4 is 2 n 2k k .For k odd, it is not hard to check that the upper bound is less than 1 + 4 Bounding σ(n, k) The goal of this section is to prove Theorem 1.6.

Lower Bound on σ(n, k)
For our proof of the lower bound in Theorem 1.6 we need the following counting lemma.
For each i, let S i be the collection of sets comparable to F i .For ease of notation, let G := {n − ℓ + 1, . . ., n} Observe that since Note that for each i > 1, we have Similarly, observe that for each i > 1, we have The final term occurs as the sets F i are counted both in their downset and their upset.Simplifying we get We now prove the lower bound given in Theorem 1.6.We actually prove a slightly stronger statement.
Note added before submission: In the final stages of preparation of this article, we noticed a recent paper of Gowty, Horsley, and Mammoliti [9], concerning the comparability number.They give a very different proof of Theorem 2.3 (see Corollary 1.2 of [9]) and use it as we do to deduce Theorem 2.4.They also provide some very interesting further analysis of the comparability number and sets that minimise c(n, m).

3 and 1. 4 . 2 k for k even and less than 1 + 1 k e 2 kConjecture 5 . 1 .
Comparing these bounds shows that they differ by a factor of e for k odd.It would be interesting to tighten this gap.We believe that (for large n) the bound given in Lemma 3.1 ought to be essentially best possible.Let k ≥ 2 be fixed and n be sufficiently large with respect to k. Thenπ(n, k) = (1 + o(1)) (k − 1) k−1 k k 2 n k .