Concerning Two Conjectures of Frick and Jafari

TL;DR

This paper advances the understanding of hypergraph chromatic numbers by improving counterexamples to a conjecture of Frick and providing new counterexamples to Jafari's conjecture, highlighting limitations in existing bounds.

Contribution

It refines previous counterexamples to Frick's conjecture and introduces several counterexamples to Jafari's conjecture, challenging their proposed inequalities.

Findings

Improved counterexamples to Frick's conjecture.

Multiple counterexamples to Jafari's conjecture.

Demonstrates limitations of current hypergraph chromatic bounds.

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 $$ \chi \left({\rm KG}^r ({\cal G}_{s-{\rm stable}})\right) \geq \left\lceil \frac{{\rm ecd}^s ({\cal G})}{r-1} \right\rceil .$$