Note on the union-closed sets conjecture and Reimer's average set size theorem

TL;DR

This paper explores the relationship between Reimer's average set size theorem and the Union-Closed Sets Conjecture, presenting methods to construct counterexamples and analyzing their properties in relation to union-closed families.

Contribution

It introduces a general method to construct infinitely many counterexamples to the implication from Reimer's conditions to the abundance condition, and analyzes their properties.

Findings

Counterexamples can be constructed with any fixed lower bound on set size.

Counterexamples can be far from being union-closed.

The paper provides a framework for understanding the gap between Reimer's conditions and the union-closed property.

Abstract

The Union-Closed Sets Conjecture, often attributed to P\'eter Frankl in 1979, remains an open problem in discrete mathematics. It posits that for any finite family of sets $S\neq\{\emptyset\}$, if the union of any two sets in the family is also in the family, then $\underline{\text{there must exist an element that belongs to at least half of the member sets}}$. We will refer to the underlined text as the abundance condition. In 2001, David Reimer proved that the average set size of a union-closed family $S$ must be at least $\frac{1}{2}\log_{2}|S|$. When proving this result, he showed that a family being union-closed implies that the family satisfies certain conditions, which we will refer to as the Reimer's conditions. Therefore, as seen in the context of Tim Gowers' polymath project on the Union-Closed Sets Conjecture, it is natural to ask if all families that satisfy Reimer's conditions meet the abundance condition. A minimal counterexample to this question was offered by Raz in 2017. In this paper, we will discuss a general method to construct infinitely many such counterexamples with any fixed lower bound on the size of the member sets. Furthermore, we will discuss some properties related to these counterexamples, especially those focusing on how far these counterexamples are from being union-closed.