Cross-Intersecting Families of Vectors

Given a sequence of positive integers (Formula presented.), let (Formula presented.) denote the family of all sequences of positive integers (Formula presented.) such that (Formula presented.) for all (Formula presented.). Two families of sequences (or vectors), (Formula presented.), are said to be (Formula presented.)-cross-intersecting if no matter how we select (Formula presented.) and (Formula presented.), there are at least (Formula presented.) distinct indices (Formula presented.) such that (Formula presented.). We determine the maximum value of (Formula presented.) over all pairs of (Formula presented.)-cross-intersecting families and characterize the extremal pairs for (Formula presented.), provided that (Formula presented.). The case (Formula presented.) 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. © 2015 Springer Japa