A counterexample to a conjecture on the chromatic number of r-stable Kneser hypergraphs

TL;DR

This paper provides a counterexample to a conjecture relating the chromatic number of r-stable Kneser hypergraphs to the colorability defect, challenging a previously believed lower bound.

Contribution

The authors construct a specific counterexample disproving the conjecture on the chromatic number of r-stable Kneser hypergraphs.

Findings

Counterexample invalidates the conjecture.

The chromatic number can be lower than the conjectured bound.

Implications for the theory of hypergraph colorings.

Abstract

The main purpose of this note is to give a counterexample to the following conjecture, raised by Florian Frick [\textit{Int. Math. Res. Not. IMRN 2020 (13), 4037-4061 (2020)}]. Conjecture. Let $r\geq 3$ and let $\mathcal{F}$ be a set system. Then $$\chi\left(\textrm{KG}^r\left(\mathcal{F}_{r-stab}\right)\right)\geq\left\lceil\frac{cd_r\left(\mathcal{F}\right)}{r-1}\right\rceil.$$