An averaging result for union-closed families of sets

Abstract

Let $\mathcal{A}$ be a union-closed family of sets with base set $b(\mathcal{A})={\bigcup }_{A\in \mathcal{A}}A$ denoted by $[n]=\{1,\cdots ,n\}$, and for any real $x>0$, let ${\mathcal{A}}_{<x}=\{A\in \mathcal{A}||A|<x\}$. Also, denote by $\mathcal{B}$ any smallest irredundant subfamily of ${\mathcal{A}}_{<n/2}$ such that $b(\mathcal{B})=b({\mathcal{A}}_{<n/2})$. We prove that if $\mathcal{A}$ is separating with height $h=4\le n$ and $0\le |\mathcal{B}|\le 2$, then the average size of a member set from $\mathcal{A}$ is at least $n/2$. We show that $h=4$ is greatest possible with respect to this result, and conclude by considering the remaining domain $3\le |\mathcal{B}|\le 4$.