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arXiv · 2507.11008

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

Ze-Chun Hu·Yi-Ding Shi·Qian-Qian Zhou
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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.

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Ze-Chun Hu, Yi-Ding Shi, Qian-Qian Zhou. 2025-07-15. A lemma on a finite union-closed family of finite sets and its applications. https://arxiv.org/abs/2507.11008

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