A constant lower bound for the union-closed sets conjecture

We show that for any union-closed family $\mathcal{F}\subseteq 2^{[n]},\mathcal{F}\mathrm{\ne }\{\mathrm{\varnothing }\}$, 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 $\mathrm{\Omega }({\log }_2(\mathrm{\mid }\mathcal{F}\mathrm{\mid }{)}^{-1})$ bounds of Knill and Wójick. 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 $H(A\cup B)>H(A)$.