Cross-intersecting families of vectors

Abstract

Given a sequence of positive integers $p=(p_1,...,p_n)$, let $S_p$ denote the family of all sequences of positive integers $x=(x_1,...,x_n)$ such that $x_i\le p_i$ for all $i$. Two families of sequences (or vectors), $A,B\subseteq S_p$, are said to be $r$-cross-intersecting if no matter how we select $x\in A$ and $y\in B$, there are at least $r$ distinct indices $i$ such that $x_i=y_i$. We determine the maximum value of $|A|\cdot |B|$ over all pairs of $r$- cross-intersecting families and characterize the extremal pairs for $r\ge 1$, provided that $\min p_i>r+1$. The case $\min p_i\le r+1$ 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, Frankl, Füredi, Livingston, Moon, and Tokushige, and answers a question of Zhang.