  We consider geometric hypergraphs whose vertex set is a finite set of points
(e.g., in the plane), and whose hyperedges are the intersections of this set
with a family of geometric regions (e.g., axis-parallel rectangles). A typical
coloring problem for such geometric hypergraphs asks, given an integer $k$, for
the existence of an integer $m=m(k)$, such that every set of points can be
$k$-colored such that every hyperedge of size at least $m$ contains points of
different (or all $k$) colors. We generalize this notion by introducing
coloring of \emph{$t$-subsets} of points such that every hyperedge that
contains enough points contains $t$-subsets of different (or all) colors. In
particular, we consider all $t$-subsets and $t$-subsets that are themselves
hyperedges. The latter, with $t=2$, is equivalent to coloring the edges of the
so-called \emph{Delaunay-graph}. In this paper we study colorings of
Delaunay-edges with respect to halfplanes, pseudo-disks, axis-parallel and
bottomless rectangles, and also discuss colorings of $t$-subsets of geometric
and abstract hypergraphs, and connections between the standard coloring of
vertices and coloring of $t$-subsets of vertices.
