# 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} }

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

#### Paper Mentions

#### 4 Citations

The Linear Relaxation of an Integer Program for the Union-Closed Conjecture

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- 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 $f(n,a)$ be… Expand

Characterizing 3-sets in Union-Closed Families

- Computer Science, Mathematics
- ArXiv
- 2019

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

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- 2017

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 Theorem… Expand

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

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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