On a restricted cross-intersection problem

Abstract

Suppose $\mathcal{A}$ and $\mathcal{B}$ are families of subsets of an n-element set and L is a set of s numbers. We say that the pair $(\mathcal{A},\mathcal{B})$ is L-cross-intersecting if $|A\cap B|\in L$ for every $A\in \mathcal{A}$ and $B\in \mathcal{B}$. Among such pairs $(\mathcal{A},\mathcal{B})$ we write $P_L(n)$ for the maximum possible value of $|\mathcal{A}||\mathcal{B}|$. In this paper we find an exact bound for $P_L(n)$ when n is sufficiently large, improving earlier work of Sgall. We also determine $P_{\{2\}}(n)$ and $P_{\{1,2\}}(n)$ exactly, which respectively confirm special cases of a conjecture of Ahlswede, Cai and Zhang and a conjecture of Sgall.