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 \(1\le s \le d\) .
