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

arXiv commentThis is a very preliminary draft