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Notes on the Union Closed Sets Conjecture
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The Union Closed Sets Conjecture states that in every finite, nontrivial set family closed under taking unions there is an element contained in at least half of all the sets of the family. We investigate two new directions with respect to the conjecture. Firstly, we investigate the frequencies of all elements among a union closed family and pose a question generalizing the Union Closed Sets Conjecture. Secondly, we investigate structures equivalent to union closed families and obtain a weakening of the Union Closed Sets Conjecture. We pose some new open questions about union closed families and related structures and hint at some further directions of research regarding the 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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A lemma on a finite union-closed family of finite sets and its applications
A lemma bounding element frequencies under deletion implies the equivalence of Frankl's conjecture and Nagel's conjecture, and strengthens a bound of Nagel for sets of size at least two.
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Further analysis on the second frequency of union-closed set families
The paper proves that if a union-closed family has second-most-frequent element frequency at most 1/3, then it must have between 81 and 113 sets and all its minimal 2-good sets have size 4.
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