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An improved lower bound for the union-closed set conjecture
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abstract
Gilmer has recently shown that in any nonempty union-closed family $\mathcal F$ of subsets of a finite set, there exists an element contained in at least a proportion $.01$ of the sets of $\mathcal F$. We improve the proportion from $.01$ to $\frac{ 3 -\sqrt{5}}{2} \approx .38$ in this result. An improvement to $\frac{1}{2}$ would be the Frankl union-closed set conjecture. We follow Gilmer's method, replacing one key estimate by a sharp estimate. We then suggest a new addition to this method and sketch a proof that it can obtain a constant strictly greater than $\frac{ 3 -\sqrt{5}}{2} $. We also disprove a conjecture of Gilmer that would have implied the union-closed set conjecture.
Forward citations
Cited by 3 Pith papers
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Frequent elements in union-closed set families
The k-th most frequent element in any union-closed set family appears in at least 1/(2^{k-1}+1) of the sets, with equality only for the near-k-cube families.
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Entropy approach for a generalization of Frankl's conjecture
A set family has an element in at least half its sets if and only if there exists an auxiliary family G satisfying an entropy inequality, giving a new equivalent form of Frankl's conjecture.
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Entropy methods in combinatorics
A selective survey of entropy methods in combinatorics, detailing randomized chain rules, Shearer's inequality, random homomorphisms, Pinsker-type arguments, the union-closed sets breakthrough, and entropy approaches ...
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