Cross-Intersecting Families of Vectors

Given a sequence of positive integers p = (p1, . . . , pn), let Sp denote the family of all sequences of positive integers x = (x1, . . . , xn) such that xi ≤ pi for all i. Two families of sequences (or vectors), A,B ⊆ Sp, are said to be r-cross-intersecting if no matter how we select x ∈ A and y ∈ B, there are at least r distinct indices i such that xi = yi. We determine the maximum value of |A| · |B| over all pairs of rcross-intersecting families and characterize the extremal pairs for r ≥ 1, provided that min pi > r + 1. The case min pi ≤ r + 1 is quite different. For this case, we have a conjecture, which we can verify under additional assumptions. Our results generalize and strengthen several previous results by Berge, Borg, Frankl, Füredi, Livingston, Moon, and Tokushige, and answers a question of Zhang.