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

12/30/2020 βˆ™ by Peter Borg, et al. βˆ™ 0 βˆ™

A set of sets is called a family. Two families π’œ and ℬ of sets are said to be cross-intersecting if each member of π’œ 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 π’œ is a non-empty subfamily of [n] ≀ r, ℬ is a non-empty subfamily of [n] ≀ s, r ≀ s, and π’œ and ℬ are cross-intersecting, then |π’œ| + |ℬ| ≀ 1 + βˆ‘_i=1^s (n i - n-r i), and equality holds if π’œ = {[r]} and ℬ is the family of sets in [n] ≀ s that intersect [r].

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