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

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 $r\ge 3$ and let $\mathcal{F}$ be a set system. Then $$\chi \left({\text{KG}}^r\left({\mathcal{F}}_{r-stab}\right)\right)\ge \lceil \frac{c{d}_r\left(\mathcal{F}\right)}{r-1}\rceil .$$