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

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 $1\leq s \leq d$.