FC-families, and improved bounds for Frankl's Conjecture

TL;DR

This paper proves Frankl's union-closed sets conjecture for specific small set families and extends bounds for the conjecture in cases where the largest set has at most nine elements.

Contribution

It establishes the conjecture for new classes of small set families and improves bounds for larger families, extending previous results by Poonen, Vaughan, Gao, and Yu.

Findings

Proved the conjecture for families with three 3-subsets of a 5-set.

Extended the conjecture's validity to four 3-subsets of a 6-set.

Confirmed the conjecture when the largest set has at most nine elements.

Abstract

A family of sets F is said to be union-closed if A \cup B is in F for every A and B in F. Frankl's conjecture states that given any finite union-closed family of sets, not all empty, there exists an element contained in at least half of the sets. Here we prove that the conjecture holds for families containing three 3-subsets of a 5-set, four 3-subsets of a 6-set, or eight 4-subsets of a 6-set, extending work of Poonen and Vaughan. As an application we prove the conjecture in the case that the largest set has at most nine elements, extending a result of Gao and Yu. We also pose several open questions.