On a restricted cross-intersection problem

Suppose A and B are families of subsets of an n-element set and L is a set of s numbers. We say that the pair (A,B) is L-cross-intersecting if |A∩B|∈L for every A∈A and B∈B. Among such pairs (A,B) we write PL(n) for the maximum possible value of |A||B|. In this paper we find an exact bound for PL(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.