A Note on a Recent Attempt to Improve the Pin-Frankl Bound

TL;DR

This paper presents a counterexample challenging a lemma in a recent attempt to improve the Pin-Frankl bound for synchronizing automata, highlighting the need for further research to resolve the open question and potentially enhance the bound.

Contribution

It identifies a flaw in a recent proof attempt and formulates an open question to guide future work on improving the Pin-Frankl bound.

Findings

Counterexample disproves a key lemma

Open question proposed to address the proof gap

Potential pathway to improve the bound

Abstract

We provide a counterexample to a lemma used in a recent tentative improvement of the the Pin-Frankl bound for synchronizing automata. This example naturally leads us to formulate an open question, whose answer could fix the line of proof, and improve the bound.