The union-closed sets conjecture almost holds for almost all random bipartite graphs
Combinatorics
2013-03-01 v1
Abstract
Frankl's union-closed sets conjecture states that in every finite union-closed set of sets, there is an element that is contained in at least half of the member-sets (provided there are at least two members). The conjecture has an equivalent formulation in terms of graphs: In every bipartite graph with least one edge, both colour classes contain a vertex belonging to at most half of the maximal stable sets. We prove that, for every fixed edge-probability, almost every random bipartite graph almost satisfies Frankl's conjecture.
Cite
@article{arxiv.1302.7141,
title = {The union-closed sets conjecture almost holds for almost all random bipartite graphs},
author = {Henning Bruhn and Oliver Schaudt},
journal= {arXiv preprint arXiv:1302.7141},
year = {2013}
}