Families of finite sets satisfying an intersection condition

Abstract

The following theorem is proved.Let X be a finite set of cardinality n ≥ 2, and let F be a family of subsets of X. Suppose that for F1, F2, F3 ∈ F we have |F1 ∩ F2 ∩ F3| ≥ 2. Then |F| ≤ 2n−2with equality holding if and only if for two different elements x, y of X, F = {F ⊆ X | x ∈ F, y ∈ F}.