# Note on the union-closed sets conjecture

## TL;DR

This paper investigates the union-closed sets conjecture, demonstrating that Reimer's condition alone does not guarantee the existence of an element present in at least half of the sets, thus clarifying limitations of previous approaches.

## Contribution

The paper shows that Reimer's condition is insufficient to prove the union-closed sets conjecture, addressing a question from Gowers' polymath project.

## Key findings

- Reimer's condition alone does not imply the conjecture.
- Clarifies limitations of previous bounds on union-closed families.
- Provides insight into the structure of union-closed set families.

## Abstract

The union-closed sets conjecture states that if a family of sets $\mathcal{A} \neq \{\emptyset\}$ is union-closed, then there is an element which belongs to at least half the sets in $\mathcal{A}$. In 2001, D. Reimer showed that the average set size of a union-closed family, $\mathcal{A}$, is at least $\frac{1}{2} \log_2 |\mathcal{A}|$. In order to do so, he showed that all union-closed families satisfy a particular condition, which in turn implies the preceding bound. Here, answering a question raised in the context of T. Gowers' polymath project on the union-closed sets conjecture, we show that Reimer's condition alone is not enough to imply that there is an element in at least half the sets.