Disjoint Union-Free Designs with Block Size Three

Abstract

Let n ⩾ k be positive integers. A famous question of Erdos asks for the largest size of a family F of k-subsets of an n-set such that there are no distinct A, B, C, D, ∈ F with A ∩ B = C ∩ D = 𝜑 and A ∪ B = C ∪ D. In the case k = 3, Fiiredi has conjectured that for sufficiently large n, lFl ⩽(n/2) and has constructed a family of examples achieving equality in which F is the block set of a design. Here, we characterize the designs meeting this conjectured bound.