The maximum sum and product of sizes of cross-intersecting families

Abstract

A family 𝒜 of sets is said to be t-intersecting if any two distinct sets in 𝒜 have at least t common elements. Families 𝒜i, 𝒜2, . . . , 𝒜k are said to be cross-t-intersecting if for any i and j in {1, 2, . . . , k} with , any set in intersects any set in 𝒜j on at least t elements. We present the following result: For any finite family ℱ that has at least one set of size at least t, there exists an integer k0 < | ℱ |such that for any k > k0, both the sum and product of sizes of k cross-t-intersecting sub-families 𝒜1, 𝒜2, . . ., 𝒜k (not necessarily distinct or non-empty) of ℱ are maxima if 𝒜1 = 𝒜2 = . . . = 𝒜k = Lfor some largest t-intersecting sub-family L of ℱ. We also prove that if t=1 and ℱ is the family of all subsets of a set X, then the result holds with k0 = 2 and L consisting of all subsets of X which contain a fixed element of X.