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

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

2007-02-13

arxiv.org/abs/math/0702348v1

European Journal of Combinatorics 27 (2006) 269-282

Mathematics

Combinatorics

19 pgs, no figures

Scientific paper

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.

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