  We prove lower and upper bounds for the chromatic number of certain
hypergraphs defined by geometric regions. This problem has close relations to
conflict-free colorings. One of the most interesting type of regions to
consider for this problem is that of the axis-parallel rectangles. We
completely solve the problem for a special case of them, for bottomless
rectangles. We also give an almost complete answer for half-planes and pose
several open problems. Moreover we give efficient coloring algorithms.
