# 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}
}
• Published 2015
• Mathematics, Computer Science
• Graphs and Combinatorics
We survey the state of the union-closed sets conjecture.
Proof of Union- Closed Sets Conjecture
We prove Union- Closed sets conjecture.
On the Union-Closed Set Conjecture
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 elementExpand
An Asymptotic Version of Frankl’s Conjecture
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
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 justifyExpand
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 thatExpand
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 supersolvableExpand
An asymptotic version of the union-closed sets conjecture
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 sameExpand
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