Graph generated union-closed families of sets
Emanuel H. Knill · arXiv (Cornell University) · 1994
Let G be a graph with vertices V and edges E. Let F be the union-closed family of sets generated by E. Then F is the family of subsets of V without isolated points. Theorem: There is an edge e belongs to E such that |{U belongs to F | e belongs to U}| = 1/2|H|. This is a special case of the union-closed sets conjecture. To put this result in perspective, a brief overview of research on the union-closed sets conjecture is given. A proof of a strong version of the theorem on graph-generated families of sets is presented. This proof depends on an analysis of the local properties of F and an application of Kleitman's lemma. Much of the proof applies to arbitrary union-closed families and can be used to obtain bounds on |{U belongs to F | e belongs to U}|/|F|.