Notes on the Union Closed Sets Conjecture

TL;DR

This paper explores new directions and structural aspects of the Union Closed Sets Conjecture, proposing generalized questions and weaker forms, and highlighting open problems for future research.

Contribution

It introduces generalized questions about element frequencies and weaker conjectures related to union closed families, expanding the theoretical understanding of the conjecture.

Findings

Proposes a generalization of the Union Closed Sets Conjecture.

Identifies structures equivalent to union closed families.

Poses new open questions and research directions.

Abstract

The Union Closed Sets Conjecture states that in every finite, nontrivial set family closed under taking unions there is an element contained in at least half of all the sets of the family. We investigate two new directions with respect to the conjecture. Firstly, we investigate the frequencies of all elements among a union closed family and pose a question generalizing the Union Closed Sets Conjecture. Secondly, we investigate structures equivalent to union closed families and obtain a weakening of the Union Closed Sets Conjecture. We pose some new open questions about union closed families and related structures and hint at some further directions of research regarding the conjecture.