# New Conjectures for Union-Closed Families

@article{Pulaj2016NewCF,
title={New Conjectures for Union-Closed Families},
author={Jonad Pulaj and Annie Raymond and Dirk Oliver Theis},
journal={Electron. J. Comb.},
year={2016},
volume={23},
pages={P3.23}
}
• Published 30 November 2015
• Mathematics, Computer Science
• Electron. J. Comb.
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 $a,n\in \mathbb{N}^+$. Similarly, one could prove that the minimum number of sets containing the… Expand
4 Citations

#### Paper Mentions

The Linear Relaxation of an Integer Program for the Union-Closed Conjecture
• Mathematics
• 2020
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. Let $f(n,a)$ beExpand
Characterizing 3-sets in Union-Closed Families
Poonen's Theorem characterizes the existence of weights on $[n]$ which ensure all UC families that contain $\mathcal{A}$ satisfy Frankl's conjecture, however the determination of such weights for specific $A$ is nontrivial even for small $n$. Expand
Cutting Planes for Union-Closed Families
Frankl’s (union-closed sets) conjecture states that for any nonempty finite union-closed (UC) family of distinct sets there exists an element in at least half of the sets. Poonen’s TheoremExpand
Cutting planes for families implying Frankl's conjecture
• Mathematics, Computer Science
• Math. Comput.
• 2020
A cutting-plane method is designed that computes the explicit weights which imply the existence conditions of Poonen’s Theorem and allows us to find a counterexample to a ten-year-old conjecture by R. Morris about the structure of generators for Non–FC-families. Expand

#### References

SHOWING 1-10 OF 51 REFERENCES
A Stability Result for the Union-Closed Size Problem
• Tom Eccles
• Mathematics, Computer Science
• Combinatorics, Probability and Computing
• 2015
The union-closed conjecture is proved for families of at least ( $\tfrac{2}{3}$ − c)2 n sets, for a positive constant c. Expand
On the Scope of Averaging for Frankl’s Conjecture
• Mathematics, Computer Science
• Order
• 2009
Let $\mathcal F$ be a union-closed family of subsets of an m-element set A. Let $n=|{\mathcal F}|\ge 2$. For b ∈ A let w(b) denote the number of sets in $\mathcal F$ containing b minus the number ofExpand
On averaging Frankl's conjecture for large union-closed-sets
• G. Czédli
• Computer Science, Mathematics
• J. Comb. Theory, Ser. A
• 2009
The sum of the n-2s(a), for all a@?A, is shown to be non-positive and this stronger version does not hold for all union-closed families; however, it is conjecture that it holds for a much wider class of families than considered here. Expand
FC-families and improved bounds for Frankl's conjecture
• R. Morris
• Mathematics, Computer Science
• Eur. J. Comb.
• 2006
Frankl's conjecture is proved in the case that the largest set has at most nine elements, extending a result of Gao and Yu and posing several open questions. 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
Formalizing Frankl's Conjecture: FC-Families
• Mathematics, Computer Science
• AISC/MKM/Calculemus
• 2012
A formalization of the computer assisted approach for proving that a family is an FC-family, where proof assistant Isabelle/HOL is used both to check mathematical content, and to perform (verified) combinatorial searches on which the proofs rely. Expand
Minimal Weight in Union-Closed Families
This paper shows how Reimer's bound may be improved if some additional information about the domain $\Omega$ of $\mathcal{S}$ is added, and gives a lower bound on the average degree. Expand
The 11-element case of Frankl's conjecture
• Mathematics, Computer Science
• Electron. J. Comb.
• 2008
In 1979, P. Frankl conjectured that in a finite union-closed family of finite sets, there has to be an element that belongs to at least half of the sets in ${\cal F}$. Expand
The union-closed sets conjecture almost holds for almost all random bipartite graphs
• Computer Science, Mathematics
• Eur. J. Comb.
• 2017
It is proved that, for every fixed edge-probability, almost every random bipartite graph almost satisfies Frankls conjecture. Expand
On the Union-Closed Sets Conjecture
Here are some notation and convention that we will adopt in this article: We use abbreviated notation for collections of sets of integers. For example, {{1, 2}, {1, 2, 3}, {3, 4}} denoted by {12,Expand