A set of sets is called a family. Two families A and B 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 <= k <= n, let ([n]<=k) denote the family of all subsets of {1, . . . , n} that have at most k elements. We show that if A is a subfamily of ([m]<=r), B is a subfamily of ([n]<=s), and A and B are cross-intersecting, then |A||B|…