The Journey of the Union-Closed Sets Conjecture

Bruhn, Henning; Schaudt, Oliver

doi:10.1007/s00373-014-1515-0

Cited by 58 publications

(56 citation statements)

References 46 publications

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“…be defined as the set of integer vectors contained in the polyhedron defined by Inequalities (2), (4) and (5).…”

Section: Proposition 4 Let a Be A Uc Family Such That ∅ ∈ A And Letmentioning

confidence: 99%

“…Yet, without witnessing first-hand computations for fixed UC families A such that U (A) = [n] and 6 ≤ n ≤ 10, Algorithm 1 may appear fraught with theoretical dangers. 5 However, in practice our method is well-behaved in the described range, and is consequently the currently best available technique for the exact determination of FC-families.…”

Section: Relaxation Questionsmentioning

confidence: 99%

“…In this paper we focus on a well-established method employed to attack the problem referred to as local configurations in Bruhn and Schaudt [5], namely UC families that imply the conjecture for all UC families which contain them.…”

Section: Introductionmentioning

confidence: 99%

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Cutting planes for families implying Frankl’s conjecture

Pulaj

2019

Math. Comp.

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We find previously unknown families of sets which ensure Frankl's conjecture holds for all families that contain them using an algorithmic framework. The conjecture states that for any nonempty union-closed (UC) family there exists an element of the ground set in at least half the sets of the considered UC family. Poonen's Theorem characterizes the existence of weights which determine whether a given UC family implies the conjecture for all UC families which contain it. We design a cutting-plane method that computes the explicit weights which imply the existence conditions of Poonen's Theorem. This method enables us to answer several open questions regarding structural properties of UC families, including the construction of a counterexample to a conjecture of Morris from 2006.

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“…be defined as the set of integer vectors contained in the polyhedron defined by Inequalities (2), (4) and (5).…”

Section: Proposition 4 Let a Be A Uc Family Such That ∅ ∈ A And Letmentioning

confidence: 99%

Section: Relaxation Questionsmentioning

confidence: 99%

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Cutting planes for families implying Frankl’s conjecture

Pulaj

2019

Math. Comp.

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“…Many other results have been discovered throughout the years. For a more complete history of the problem, we refer the reader to the following excellent survey [BS14]. Finally, we note that Timothy Gowers recently led a polymath project, FUNC, on this topic.…”

Section: Introductionmentioning

confidence: 99%

New Conjectures for Union-Closed Families

Pulaj

Raymond

Theis

2016

Electron. J. Combin.

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The Frankl conjecture, also known as the union-closed sets conjecture, states that in any finite non-empty union-closed family, there exists an element in at least half of the sets. From an optimization point of view, one could instead prove that 2a is an upper bound to the number of sets in a union-closed family on a ground set of n elements where each element is in at most a sets for all a, n ∈ N + . Similarly, one could prove that the minimum number of sets containing the most frequent element in a (non-empty) unionclosed family with m sets and n elements is at least m 2 for any m, n ∈ N + . Formulating these problems as integer programs, we observe that the optimal values we computed do not vary with n. We formalize these observations as conjectures, and show that they are not equivalent to the Frankl conjecture while still having wide-reaching implications if proven true. Finally, we prove special cases of the new conjectures and discuss possible approaches to solve them completely.

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“…Possibly as a reflection of its general difficulty, Gowers [9] suggested that work on this conjecture could fruitfully be done as a collaborative Polymath project. See [2] for a survey of the literature on the union-closed sets conjecture.…”

Section: Introductionmentioning

confidence: 99%

The graph formulation of the union-closed sets conjecture

Bruhn

Charbit

Schaudt

et al. 2015

European Journal of Combinatorics

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International audienceThe union-closed sets conjecture asserts that in a finite non-trivial union-closed family of sets there has to be an element that belongs to at least half the sets. We show that this is equivalent to the conjecture that in a finite non-trivial graph there are two adjacent vertices each belonging to at most half of the maximal stable sets. In this graph formulation other special cases become natural. The conjecture is trivially true for non-bipartite graphs and we show that it holds also for the classes of chordal bipartite graphs, subcubic bipartite graphs, bipartite series-parallel graphs and bipartitioned circular interval graphs. We derive that the union-closed sets conjecture holds for all union-closed families being the union-closure of sets of size at most three

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The Journey of the Union-Closed Sets Conjecture

Abstract: We survey the state of the union-closed sets conjecture.

Cited by 58 publications

References 46 publications

Cutting planes for families implying Frankl’s conjecture

Cutting planes for families implying Frankl’s conjecture

New Conjectures for Union-Closed Families

The graph formulation of the union-closed sets conjecture

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