New Conjectures for UnionClosed Families
Abstract
The Frankl conjecture, also known as the unionclosed sets conjecture, states that in any finite nonempty unionclosed 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 unionclosed 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 most frequent element in a (nonempty) unionclosed family with $m$ sets and $n$ elements is at least $\frac{m}{2}$ for any $m,n\in \mathbb{N}^+$. 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 widereaching implications if proven true. Finally, we prove special cases of the new conjectures and discuss possible approaches to solve them completely.
 Publication:

arXiv eprints
 Pub Date:
 November 2015
 arXiv:
 arXiv:1512.00083
 Bibcode:
 2015arXiv151200083P
 Keywords:

 Mathematics  Combinatorics;
 Mathematics  Optimization and Control
 EPrint:
 16 pages