The maximum sum of sizes of cross-intersecting families of subsets of a set

A set of sets is called a family. Two families 𝒜 and ℬ of sets are said to be cross-intersecting if each member of 𝒜 intersects each member of ℬ. For any two integers n and k with 1 ≤ k ≤ n, let [n] ≤ k denote the family of subsets of [n] = {1, …, n} that have at most k elements. We show that if 𝒜 is a non-empty subfamily of [n] ≤ r, ℬ is a non-empty subfamily of [n] ≤ s, r ≤ s, and 𝒜 and ℬ are cross-intersecting, then |𝒜| + |ℬ| ≤ 1 + ∑_i=1^s (n i - n-r i), and equality holds if 𝒜 = {[r]} and ℬ is the family of sets in [n] ≤ s that intersect [r].

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