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Better bounds for the union-closed sets conjecture using the entropy approach
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abstract
We improve the best known constant $\frac{3-\sqrt 5}{2}$ for which the union-closed conjecture is known to be true, by using dependent samples as suggested by Sawin and the entropy approach on this problem initiated by Gilmer. Meanwhile, we focus on the intuition behind this entropy approach and its boundaries.
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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