# The Journey of the Union-Closed Sets Conjecture

@article{Bruhn2015TheJO, title={The Journey of the Union-Closed Sets Conjecture}, author={Henning Bruhn and Oliver Schaudt}, journal={Graphs and Combinatorics}, year={2015}, volume={31}, pages={2043-2074} }

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

#### 44 Citations

On the Union-Closed Set Conjecture

- Mathematics
- 2016

We provide a simple proof for the union-closed sets conjecture, a long-standing open problem in set theory with immediate applications to graph theory, number theory, and order-theory.

A Stronger Version of the Union-closed Sets Conjecture

- Mathematics
- 2017

The union-closed sets conjecture (Frankl's conjecture) says that for any finite union-closed family of finite sets, other than the family consisting only of the empty set, there exists an element… Expand

An Asymptotic Version of Frankl’s Conjecture

- Computer Science
- Am. Math. Mon.
- 2021

It is shown that the maximum average set size in complements of union-closed families over n elements is . Expand

A Note on the Frankl Conjecture

- Mathematics
- 2019

The Frankl conjecture (called also union-closed sets conjecture) is one of the famous unsolved conjectures in combinatorics of finite sets. In this short note, we introduce and to some extent justify… Expand

Two Stronger Versions of the Union-closed Sets Conjecture

- Mathematics
- 2017

The union-closed sets conjecture (Frankl’s conjecture) says that for any finite unionclosed family of finite sets, other than the family consisting only of the empty set, there exists an element that… Expand

The graph formulation of the union-closed sets conjecture

- Mathematics, Computer Science
- Eur. J. Comb.
- 2015

The conjecture 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 is trivially true for non-bipartite graphs and it holds also for the classes of chordal bipartites graphs, subcubic bipartITE graphs, bipartite series-parallel graphs and bipartitioned circular interval graphs. Expand

Frankl's Conjecture for Subgroup Lattices

- Computer Science, Mathematics
- Electron. J. Comb.
- 2017

We show that the subgroup lattice of any finite group satisfies Frankl's Union-Closed Conjecture. We show the same for all lattices with a modular coatom, a family which includes all supersolvable… Expand

An asymptotic version of the union-closed sets conjecture

- Mathematics
- 2020

We show that the biggest possible average set size in the complement $2^{\{1,2,\ldots, n\}} \setminus A$ of a union-closed family $A \subset 2^{\{1,2, \ldots, n\}}$ is $\tfrac{n+1}{2}$. With the same… Expand

Generation of Union-Closed Sets and Moore Families

- Mathematics, Computer Science
- J. Integer Seq.
- 2018

An algorithm to constructively enumerate non-isomorphic Union closed Sets and Moore sets is described and it seems unlikely that constructive enumeration for 8 or more elements will be possible in the foreseeable future. Expand

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The graph formulation of the union-closed sets conjecture

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The conjecture 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 is trivially true for non-bipartite graphs and it holds also for the classes of chordal bipartites graphs, subcubic bipartITE graphs, bipartite series-parallel graphs and bipartitioned circular interval graphs. Expand

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