Cross-intersecting families of vectors

TL;DR

This paper determines the maximum product of sizes for pairs of vector families with a cross-intersecting property, characterizes extremal pairs when the minimum sequence element exceeds a threshold, and discusses a conjecture for the other case.

Contribution

It generalizes and strengthens previous results on cross-intersecting families of vectors, providing a complete characterization for the case when the minimum element exceeds a certain bound.

Findings

Maximum product of sizes for r-cross-intersecting families determined.

Extremal pairs characterized for the case   p_i > r+1.

Conjecture proposed and verified under additional assumptions for the other case.

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, F\"uredi, Livingston, Moon, and Tokushige, and answers a question of Zhang.