A lemma on a finite union-closed family of finite sets and its applications

TL;DR

This paper proves a lemma about the structure of finite union-closed families of sets, relating the proportions of certain subsets, and explores its applications to combinatorial problems.

Contribution

It introduces a new lemma connecting subset proportions within union-closed families and demonstrates its applications in combinatorics.

Findings

Established a lower bound for the proportion of sets containing a specific element.

Provided applications demonstrating the lemma's utility in combinatorial contexts.

Enhanced understanding of the structure of union-closed families.

Abstract

Suppose that $\mathscr{F}$ is a finite union-closed family of sets with $\cup_{A\in \mathscr{F}}A=\{1,2,\ldots,m\}$ and $m\geq 2$. Fix $i\in \{1,2,\ldots,m\}$ and denote $\mathscr{G}:=\{A\backslash \{i\}: A\in \mathscr{F}\}$. For $j\in \{1,2,\ldots,m\}\backslash\{i\}$, let $\mathscr{G}_j:=\{A\in\mathscr{G}: j\in A\}$ and $\mathscr{F}_j:=\{A\in\mathscr{F}: j\in A\}$. In this note, we will prove a lemma which says that if $\frac{|\mathscr{G}_j|}{|\mathscr{G}|}\geq c\,(c\in (0,1])$, then $\frac{|\mathscr{F}_j|}{|\mathscr{F}|}\geq \frac{1}{1+2(1-c)/c}$. Several applications of this lemma will be given.