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

Abstract

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

7 pages