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 P L ( n ) for the maximum possible value of | A | | 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.