Minimum density of union-closed families

Details Minimum density of union-closed families Minimum density of union-closed families

2011-06-02

arxiv.org/abs/1106.0369v1

Mathematics

Combinatorics

Scientific paper

Let F be a finite union-closed family of sets whose largest set contains n elements. In \cite{Wojcik92}, Wojcik defined the density of F to be the ratio of the average set size of F to n and conjectured that the minimum density over all union-closed families whose largest set contains n elements is (1 + o(1))\log_2(n)/(2n) as n approaches infinity. We use a result of Reimer \cite{Reimer03} to show that the density of F is always at least log_2(n)/(2n), verifying Wojcik's conjecture. As a corollary we show that for n \geq 16, some element must appear in at least \sqrt{(\log_2(n))/n}(|F|/2) sets of F.

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