The maximum product of sizes of cross-intersecting families

Abstract

We say that a set A t-intersects a set B if A and B have at least t common elements. Two families A and B of sets are said to be cross-t-intersecting if each set in A t-intersects each set in B. A subfamily S of a family F is called a t-star of F if the sets in S have t common elements. Let l(F, t) denote the size of a largest t-star of F. We call F a (≤ r)-family if each set in F has at most r elements. We determine a function c : N 3 → N such that the following holds. If A is a subfamily of a (≤ r)-family F with l(F, t) ≥ c(r, s, t)l(F, t + 1), B is a subfamily of a (≤ s)-family G with l(G, t) ≥ c(r, s, t)l(G, t + 1), and A and B are cross-t-intersecting, then |A||B| ≤ l(F, t)l(G, t). Some known results follow from this, and we identify several natural classes of families for which the bound is attained.

Onr-crosst-intersecting families for weak compositions

“…, A r are r-cross t-intersecting if |A 1 ∩A 2 ∩· · ·∩A r | ≥ t holds for all A i ∈ A i . It has been shown by Frankl and Tokushige [10] (see [4,20,26] for other related results) that if…”

On the Erdős–Ko–Rado theorem and the Bollobás theorem fort-intersecting families

“…Let 1 ≤ t ≤ k 1 ≤ k 2 ≤ · · · ≤ k r ≤ n and n ≥ n 0 (k r−1 , k r , t) with n 0 (k r−1 , k r , t) as in Theorem 1.2 in [3]. For each F i ⊆  …”

“…, F r of subsets of [n] are cross t-intersecting if |A ∩ B| ≥ t for every A ∈ F i and B ∈ F j , where i ̸ = j. In 2014, Borg proved the following theorem [3]. Theorem 1.4 (Borg [3]).…”

The maximum product of weights of cross-intersecting families

“…The maximum product problem for [n] r was first addressed by Pyber [43], who proved that, for any r, s, and n such that either r = s n/2 or r < s and n 2s + r − 2, if A ⊆ [ [39] proved this for r s n/2 (see also [5]). It has been shown in [13] that, for t r s, there exists an integer n 0 (r, s, t) such that, for n n 0 (r, s, t), if A ⊆ [n] r , B ⊆ [n] s , and A and B are cross-t-intersecting, then |A||B|…”

“…The value of n 0 (r, s, t) given in [13] is far from best possible. The special case r = s is treated in [25,46,48], which establish values of n 0 (r, r, t) that are close to the conjectured smallest value of (t + 1)(r − t + 1), and which use algebraic methods and Frankl's random walk method [21]; in particular, n 0 (r, r, t) = (t + 1)r is determined in [25] for t 14.…”