Cross-sperner families

Abstract

A pair of families (F, G) is said to be cross-Sperner if there exists no pair of sets F ∈ F, G ∈ G with F ⊆ G or G ⊆ F . There are two ways to measure the size of the pair (F, G): with the sum |F| + |G| or with the product |F| · |G|. We show that if F, G ⊆ 2 [n] , then |F||G| ≤ 2 2n−4 and |F| + |G| is maximal if F or G consists of exactly one set of size ⌈n/2⌉ provided the size of the ground set n is large enough and both F and G are non-empty.

Improved bounds for cross-Sperner systems

“…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 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.…”

“…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 , .…”

On colorings of the Boolean lattice avoiding a rainbow copy of a poset

“…Theorem 3.1 (Ahlswede, Zhang [1], Gerbner et al [5]). If F 1 , F 2 ⊆ 2 [n] are families such that any pair…”

“…, F l with the property that for any F i ∈ F i and F j ∈ F j with i = j the sets F i and F j are incomparable. They called these families cloud antichains, later Gerbner et al [5] studied them under the name of cross-Sperner families. The upper bound on f (n, 2, P 2 ) follows from the following theorem.…”