  Let $S$ be a $k$-colored (finite) set of $n$ points in $\mathbb{R}^d$, $d\geq
3$, 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
$3 \leq k \leq d$ we provide a lower bound of $\Omega(n^{d-k+1+2^{-d}})$ and
strengthen this to $\Omega(n^{d-2/3})$ for $k=2$. On the way we provide various
results on triangulations of point sets in $\mathbb{R}^d$. In particular, for
any constant dimension $d\geq3$, we prove that every set of $n$ points ($n$
sufficiently large), in general position in $\mathbb{R}^d$, admits a
triangulation with at least $dn+\Omega(\log n)$ simplices.
