FC-families and improved bounds for Frankl’s conjecture

A family of sets AA is said to be union-closed if {A∪B:A,B∈A}⊂A{A∪B:A,B∈A}⊂A. 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.