Local Configurations in Union-Closed Families

TL;DR

This paper advances the understanding of the Frankl union-closed sets conjecture by establishing new bounds, exploring symmetry, and introducing local configuration criteria to prove the conjecture for various classes of families.

Contribution

It provides new bounds for union-closed families satisfying the conjecture, addresses symmetry questions, and generalizes Poonen's Theorem to prove the conjecture in new cases.

Findings

New bounds for $FC(k, n)$ in union-closed families.

Answering Vaughan's symmetry question.

Proving the conjecture for several previously unknown classes.

Abstract

The Frankl or Union-Closed Sets conjecture states that for any finite union-closed family of sets $\mathcal{F}$ containing some nonempty set, there is some element $i$ in the ground set $U(\mathcal F) := \bigcup_{S \in \mathcal{F}} S$ of $\mathcal{F}$ such that $i$ is in at least half of the sets in $\mathcal{F}$. In this work, we find new values and bounds for the least integer $FC(k, n)$ such that any union-closed family containing $FC(k, n)$ distinct $k$-sets of an $n$-set $X$ satisfies Frankl's conjecture with an element of $X$. Additionally, we answer an older question of Vaughan regarding symmetry in union-closed families and we give a proof of a recent question posed by Ellis, Ivan and Leader. Finally, we introduce novel local configuration criteria through a generalization of Poonen's Theorem to prove the conjecture for many, previously unknown classes of families.