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 χ L(k,d) (resp. χ PL(k,d) ) the supremum H χ (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. χ L(k,d)=∞ for all 2≤ k≤ d , B. χ PL(d+1,d)=∞ and C. χ L(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. χ ₛ(M)=∞ for 1≤ s ≤ d .