Note: a counterexample to a conjecture of Gilmer which would imply the union-closed conjecture

TL;DR

This paper provides a counterexample to Gilmer's conjecture, which if true, would have supported the union-closed conjecture, thereby challenging a potential pathway to prove the union-closed conjecture.

Contribution

It presents a specific counterexample to Gilmer's conjecture, disproving a proposed implication for the union-closed conjecture.

Findings

Counterexample invalidates Gilmer's conjecture

Disproves the implication to the union-closed conjecture

Challenges previous assumptions linking the conjecture to Gilmer's work

Abstract

In this very short note, we give a counterexample to a recent conjecture of Gilmer which would have implied the union-closed conjecture.