Local Configurations in Union-Closed Families

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.