A stability result for the union-closed size problem
A family of sets is called union-closed if whenever $A$ and $B$ are sets of the family, so is $A\cup B$. 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 least $p_02^n$ sets for some constant $p_0$. The first result in this direction appears in a recent paper of Balla, Bollob\'as and Eccles \cite{BaBoEc}, who showed that union-closed families of at least $\frac{2}{3}2^n$ sets 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 $\frac{2}{3}$. Here, we provide a stability result for the main theorem of \cite{BaBoEc}, and as a consequence we prove the union-closed conjecture for families of at least $(\frac{2}{3}-c)2^n$ sets, for a positive constant $c$.