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

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 A 1 , A 2 , … , A k are said to be cross-t-intersecting if for any i and j in { 1 , 2 , … , k } with i ≠ j , any set in A i intersects any set in A j 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 k 0 ⩽ | F | such that for any k ⩾ k 0 , both the sum and product of sizes of k cross-t-intersecting sub-families A 1 , A 2 , … , A k (not necessarily distinct or non-empty) of F are maxima if A 1 = A 2 = ⋯ = A k = L for 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 k 0 = 2 and L consisting of all subsets of X which contain a fixed element of X.