On the minimum number of disjoint pairs in a family of finite sets

Abstract

The following conjecture of Katona is proved. Let X be a finite set of cardinality n, 1 ⩽ m ⩽ 2 n . Then there is a family F , | F | = m, such that F ∈ F , G ⊆ X, | G | > | F | implies G ∈ F and F minimizes the number of pairs ( F 1, F 2), F 1, F 2 ∈ F F 1 ∩ F 2 = ⊘ over all families consisting of m subsets of X.