The maximum product of sizes of cross-intersecting families

A set of sets is called a family. Two families A and B are said to be cross-t-intersecting if each set in A intersects each set in B in at least t elements. For a family.F, let l(F, t) denote the size of a largest subfamily of F whose sets have at least t common elements. We call F a (<= r)-family if each set in F has at most r elements. We show that for any positive integers r, s and t, there exists an integer c(r, s, t) such that the following holds. If A is a subfamily of a (<= r)-family F with l(F, t) >= c(r, s, t)l(F, t + 1), B is a subfamily of a (<= s)-family g with l(g, t) >= c(r, s, t)l(g, t+ 1), and A and B are cross-t-intersecting, then |A| |B| <= l(F, t)l(g, t). We give c(r, s, t) explicitly. Some known results follow from this, and we identify several natural classes of families for which the bound is attained. (C) 2017 Elsevier B.V. All rights reserved.