New Conjectures for Union-Closed Families

TL;DR

This paper introduces new conjectures related to the union-closed sets conjecture, explores their implications, and proves some special cases, aiming to advance understanding of this longstanding combinatorial problem.

Contribution

The paper formulates new conjectures equivalent to certain bounds in union-closed families, distinguishes them from the Frankl conjecture, and proves some special cases.

Findings

Optimal values are invariant with respect to the number of elements.

New conjectures are not equivalent to the Frankl conjecture.

Some special cases of the conjectures are proven.

Abstract

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 most frequent element in a (non-empty) union-closed 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 wide-reaching implications if proven true. Finally, we prove special cases of the new conjectures and discuss possible approaches to solve them completely.