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 A1,A2,…,Ak are said to be cross-t-intersecting if for any i and j in {1,2,…,k} with i≠j, any set in Ai 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=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 k0=2 and L consisting of all subsets of X which contain a fixed element of X.