58 research outputs found
A constant lower bound for the union-closed sets conjecture
We show that for any union-closed family , there exists an which is contained
in a fraction of the sets in . This is the first known
constant lower bound, and improves upon the
bounds of Knill and W\'{o}jick. Our result
follows from an information theoretic strengthening of the conjecture.
Specifically, we show that if are independent samples from a
distribution over subsets of such that for all
and , then .Comment: 9 pages, 1 figure. (Update 11/28/22: Typos fixed, and added reference
to follow up work improving the bound and refuting Conjecture 1.
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