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

Ze-Chun Hu, Yi-Ding Shi, Qian-Qian Zhou

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 $\mathscr{…