The union-closed sets conjecture almost holds for almost all random bipartite graphs

TL;DR

This paper demonstrates that for almost all random bipartite graphs with a fixed edge probability, the union-closed sets conjecture is nearly satisfied, indicating the conjecture holds in a probabilistic sense for such graphs.

Contribution

It establishes that almost every random bipartite graph nearly satisfies Frankl's union-closed sets conjecture, providing probabilistic evidence for the conjecture's validity.

Findings

Almost all random bipartite graphs satisfy the conjecture approximately.

The conjecture holds with high probability in the random graph model.

The result links graph properties with the union-closed sets conjecture.

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.