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 r-cross-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.
ASJC Scopus subject areas
- Theoretical Computer Science
- Discrete Mathematics and Combinatorics