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arXiv · 2211.09055

A constant lower bound for the union-closed sets conjecture

Justin Gilmer
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Abstract

We show that for any union-closed family $\mathcal{F} \subseteq 2^{[n]}, \mathcal{F} \neq \{\emptyset\}$, there exists an $i \in [n]$ which is contained in a $0.01$ fraction of the sets in $\mathcal{F}$. This is the first known constant lower bound, and improves upon the $\Omega(\log_2(|\mathcal{F}|)^{-1})$ bounds of Knill and W\'{o}jick. Our result follows from an information theoretic strengthening of the conjecture. Specifically, we show that if $A, B$ are independent samples from a distribution over subsets of $[n]$ such that $Pr[i \in A] < 0.01$ for all $i$ and $H(A) > 0$, then $H(A \cup B) > H(A)$.

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Justin Gilmer. 2022-11-16. A constant lower bound for the union-closed sets conjecture. https://arxiv.org/abs/2211.09055

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