Empty Monochromatic Simplices

Let S be a k-colored (finite) set of n points in R, d ≥ 3, in general position, that is, no (d+1) points of S lie in a common (d−1)-dimensional hyperplane. We count the number of empty monochromatic d-simplices determined by S, that is, simplices which have only points from one color class of S as vertices and no points of S in their interior. For 3 ≤ k ≤ d we provide a lower bound of Ω(nd−k+1+2 −d ) and strengthen this to Ω(nd−2/3) for k = 2. On the way we provide various results on triangulations of point sets in R. In particular, for any constant dimension d ≥ 3, we prove that every set of n points (n sufficiently large), in general position in R, admits a triangulation with at least dn+ Ω(log n) simplices.