FC-families, and improved bounds for Frankl's Conjecture
Combinatorics
2007-05-23 v1
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.
Cite
@article{arxiv.math/0702348,
title = {FC-families, and improved bounds for Frankl's Conjecture},
author = {Robert Morris},
journal= {arXiv preprint arXiv:math/0702348},
year = {2007}
}
Comments
19 pgs, no figures