Concerning Two Conjectures of Frick and Jafari

Abstract

In 2022, Hamid Reza Daneshpajouh provided some counterexamples to the following conjecture of Florian Frick. \bf Conjecture. Let $r \geq 3$. Then, every hypergraph ${\cal G}$ over the ground set $[n]$ satisfies $$ \chi \left({\rm KG}^r ({\cal G}_{r-{\rm stable}})\right) \geq \left\lceil \frac{{\rm cd}^r ({\cal G})}{r-1} \right\rceil . $$ In this paper, we improve Danashpajouh's results. Also, we provide several counterexamples to the following recent conjecture of Amir Jafari. Conjecture. If $s \geq r \geq 2$, then for any hypergraph ${\cal G}$ with vertex set $[n]$ we have $$ \chi \left({\rm KG}^r ({\cal G}_{s-{\rm stable}})\right) \geq \left\lceil \frac{{\rm ecd}^s ({\cal G})}{r-1} \right\rceil .$$