On colorings of hypergraphs embeddable in $\mathbb{R}^d$

TL;DR

This paper investigates the coloring properties of hypergraphs embeddable in Euclidean space, establishing new bounds and infinite chromatic numbers for various classes of hypergraphs and their applications to triangulated manifolds.

Contribution

It improves existing results by proving infinite chromatic numbers for certain hypergraph classes and extends these findings to triangulations of manifolds.

Findings

hi_L(k,d)= hi_{PL}(d+1,d)= hi_L(d+1,d)or odd dnd hi_s(M)=or 1nd s.

Abstract

The (weak) chromatic number of a hypergraph $H$, denoted by $\chi(H)$, is the smallest number of colors required to color the vertices of $H$ so that no hyperedge of $H$ is monochromatic. For every $2\le k\le d+1$, denote by $\chi_L(k,d)$ (resp. $\chi_{PL}(k,d)$) the supremum $\sup_H \chi(H)$ where $H$ runs over all finite $k$-uniform hypergraphs such that $H$ forms the collection of maximal faces of a simplicial complex that is linearly (resp. PL) embeddable in $\mathbb{R}^d$. Following the program by Heise, Panagiotou, Pikhurko and Taraz, we improve their results as follows: For $d \geq 3$, we show that A. $\chi_L(k,d)=\infty$ for all $2\le k\le d$, B. $\chi_{PL}(d+1,d)=\infty$ and C. $\chi_L(d+1,d)\ge 3$ for all odd $d\ge 3$. As an application, we extend the results by Lutz and M{\o}ller on the weak chromatic number of the $s$-dimensional faces in the triangulations of a fixed triangulable $d$-manifold $M$: D. $\chi_s(M)=\infty$ for $1\leq s \leq d$.