On Colorings of Hypergraphs Embeddable in Rd

The (weak) chromatic number of a hypergraph H, denoted by χ(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≤k≤d+1, denote by χL(k,d) (resp. χPL(k,d)) the supremum supHχ(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 Rd. Following the program by Heise, Panagiotou, Pikhurko and Taraz, we improve their results as follows: For d≥3, we show that A. χL(k,d)=∞ for all 2≤k≤d, B. χPL(d+1,d)=∞ and C. χL(d+1,d)≥3 for all odd d≥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. χs(M)=∞ for 1≤s≤d.