The Union-Closed Sets Conjecture for Small Families

Maβberg, Jens

doi:10.1007/s00373-016-1701-3

Cited by 3 publications

(2 citation statements)

References 6 publications

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“…For more details on the development of the conjecture and what is known about it see the survey [7]. In the recent years [1,2,6,12,18,21,28,29,30] have further been published investigating the conjecture with respect to one aspect or another. In this context, the collaborative effort in [16] should also be mentioned.…”

Section: Introductionmentioning

confidence: 99%

Notes on the Union Closed Sets Conjecture

Nagel¹

2022

Preprint

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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.

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Section: Introductionmentioning

confidence: 99%

Notes on the Union Closed Sets Conjecture

Nagel¹

2022

Preprint

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“…Falgas-Ravry [7] showed that the conjecture holds for any separating union-closed family F with a universe of n elements with at most 2n elements (separating here means that no two distinct elements i, j ∈ [n] appear in exactly the same sets in F ). This was slightly improved by Maßberg [16], from 2n to 2(n + n log 2 n−log log 2 n ). Interestingly, Hu [10] proved that if this bound can be improved to (2 + c)n for some constant c > 0 , this already implies that any union-closed family F has an element appearing in at least c−2 2(c−1) |F | sets in F .…”

Section: Introductionmentioning

confidence: 99%

Two Results on Union-Closed Families

Karpas¹

2017

Preprint

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We show that there is some absolute constant c > 0, such that for any union-closed family F ⊆ 2 [n] , if |F| ≥ ( 1 2 − c)2 n , then there is some element i ∈ [n] that appears in at least half of the sets of F. We also show that for any union-closed family F ⊆ 2 [n] , the number of sets which are not in F that cover a set in F is at most 2 n−1 , and provide examples where the inequality is tight.

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An Asymptotic Version of Frankl’s Conjecture

Studer

2021

The American Mathematical Monthly

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The Union-Closed Sets Conjecture for Small Families

Abstract: We prove that the union-closed sets conjecture is true for separatingwhere m denotes the number of elements in A.

Cited by 3 publications

References 6 publications

Notes on the Union Closed Sets Conjecture

Notes on the Union Closed Sets Conjecture

Two Results on Union-Closed Families

An Asymptotic Version of Frankl’s Conjecture

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