A Sharp Bound for the Product of Weights of Cross-Intersecting Families

Two families A and B of sets are said to be cross-intersecting if each set in A intersects each set in B. For any two integers n and k with 1 6 k 6 n, let ([n] 6k ) denote the family of subsets of {1, . . . , n} of size at most k, and let Sn,k denote the family of sets in ([n] 6k ) that contain 1. The author recently showed that if A ⊆ ( [m] 6r ) , B ⊆ ( [n] 6s ) , and A and B are cross-intersecting, then |A||B| 6 |Sm,r||Sn,s|. We prove a version of this result for the more general setting of weighted sets. We show that if g : ( [m] 6r ) → R+ and h : ( [n] 6s ) → R+ are functions that obey certain conditions, A ⊆ ( [m] 6r ) , B ⊆ ( [n] 6s ) , and A and B are cross-intersecting, then ∑