An asymptotic version of the union-closed sets conjecture

Abstract

We show that the biggest possible average set size in the complement $2^{\{1,2,\dots ,n\}}\setminus A$ of a union-closed family $A\subset 2^{\{1,2,\dots ,n\}}$ is $\frac{n+1}{2}$. With the same proof we get a sharp upper bound for the average frequency in complements of union-closed families. This implies an asymptotic version of the union-closed sets conjecture, formulated in terms of complements of union-closed families.

2 pages