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 all the other sets in A . 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 , every set in A i   t -intersects every set in A j . 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 k cross- t -intersecting subfamilies A 1 , … , A k (not necessarily distinct or non-empty) of F are maxima if A 1 = ⋯ = A k = 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 A 1 = ⋯ = A k = 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.