An upper bound for union-closed family size

Abstract

Let $\mathcal{A}$ be a union-closed family of sets with universe ${\bigcup }_{A\in \mathcal{A}}A=[n]=\{1,\cdots ,n\}$ and length $\ell$. We prove that $|\mathcal{A}|\le {\sum }_{i=0}^{\ell }\left(\genfrac{}{}{0ex}{}{n}{i}\right)$, with equality if and only if $\mathcal{A}={\bigcup }_{i=0}^{\ell }\left(\genfrac{}{}{0ex}{}{[n]}{n-i}\right)$. Additionally, by showing that $|\mathcal{A}|\le \frac{{\ell }^p-1}{\ell -1}+2^n(1-2^{-\ell }{)}^p$ for any nonnegative integer $p$, we establish for all integers $1\le k\le n$ that ${\sum }_{i=0}^k\left(\genfrac{}{}{0ex}{}{n}{i}\right)\le \frac{k^{\hat{p}}-1}{k-1}+2^n(1-2^{-k}{)}^{\hat{p}}$, where $\hat{p}=\lfloor (n-k)/{\log }_2(\frac{k}{1-2^{-k}})\rfloor +1$.