A Stability Result for the Union-Closed Size Problem

Abstract

A family of sets is called union-closed if wheneverAandBare sets of the family, so isA∪B. The long-standing union-closed conjecture states that if a family of subsets of [n] is union-closed, some element appears in at least half the sets of the family. A natural weakening is that the union-closed conjecture holds for large families, that is, families consisting of at leastp02nsets for some constantp0. The first result in this direction appears in a recent paper of Balla, Bollobás and Eccles [1], who showed that union-closed families of at least$\tfrac{2}{3}$2nsets satisfy the conjecture; they proved this by determining the minimum possible average size of a set in a union-closed family of given size. However, the methods used in that paper cannot prove a better constant than$\tfrac{2}{3}$. Here, we provide a stability result for the main theorem of [1], and as a consequence we prove the union-closed conjecture for families of at least ($\tfrac{2}{3}$−c)2nsets, for a positive constantc.