Cross-intersecting non-empty uniform subfamilies of hereditary families

Abstract

Two families A and B of sets are cross-t-intersecting if each set in A intersects each set in B in at least t elements. A family H is hereditary if for each set A in H, all the subsets of A are in H. Let H(r) denote the family of r-element sets in H. We show that for any integers t, r, and s with 1 ≤ t ≤ r ≤ s, there exists an integer c(r, s, t) such that the following holds for any hereditary family H whose maximal sets are of size at least c(r, s, t). If A is a nonempty subfamily of H(r) , B is a non-empty subfamily of H(s) , A and B are cross-t-intersecting, and |A| + |B| is maximum under the given conditions, then for some set I in H with t ≤ |I| ≤ r, either A = {A ∈ H(r) : I ⊆ A} and B = {B ∈ H(s) : |B ∩ I| ≥ t}, or r = s, t < |I|, A = {A ∈ H(r) : |A ∩ I| ≥ t}, and B = {B ∈ H(s) : I ⊆ B}. We give c(r, s, t) explicitly. The result was conjectured by the author for t = 1 and generalizes well-known results for the case where H is a power set.