On Colorings of Hypergraphs Embeddable in $$\mathbb {R}^d$$

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 $$d \ge 3$$ , 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 Møller 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 $$1\le s \le d$$ .