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

We say that a set A t-intersects a set B if A and B have at least t common elements. A family A of sets is said to be t-intersecting if each set in A t-intersects any other set in A. Families A1,A2, ...,Ak are said to be cross-t-intersecting if for any i and j in {1, 2, ..., k} with i 6= j, any set in Ai t-intersects any set in Aj . We prove that for any finite family F that has at least one set of size at least t, there exists an integer κ ≤ |F| such that for any k ≥ κ, both the sum and the product of sizes of any k cross-t-intersecting subfamilies A1, ...,Ak (not necessarily distinct or non-empty) of F are maxima if A1 = ... = Ak = L for some largest t-intersecting subfamily L of F . We then study the smallest possible value of κ and investigate the case k < κ; this includes a cross-intersection result for straight lines that demonstrates that it is possible to have F and κ such that for any k < κ, the configuration A1 = ... = Ak = L is neither optimal for the sum nor optimal for the product. We also outline solutions for various important families F , and we provide solutions for the case when F is a power set.