Atoms of set systems with a fixed number of pairwise unions

Abstract

For a finite set system H with ground set X, we let H ∨ H = {A ∪ B: A, B ∈ H, A ≠ B} . An atom of H is a nonempty maximal subset C of X such that for all A ∈ H, either C ⊂ A or C ∩ A = 0. We obtain a best possible upper bound for the number of atoms determined by a set system H with ∥ H∥ = k and ∥ H ∨ H∥ = u for all integers k and u. This answers a problem posed by Sós.