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 $h = 4 \leq n$ and $0 \leq |\mathcal{B}| \leq 2$, 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 $3 \leq |\mathcal{B}| \leq 4$.