  Let n \geq l \geq 2 and q \geq 2. We consider the minimum N such that
whenever we have N points in the plane in general position and the l-subsets of
these points are colored with q colors, there is a subset S of n points all of
whose l-subsets have the same color and furthermore S is in convex position.
This combines two classical areas of intense study over the last 75 years: the
Ramsey problem for hypergraphs and the Erd\H os-Szekeres theorem on convex
configurations in the plane. For the special case l = 2, we establish a single
exponential bound on the minimum N, such that every complete $N$-vertex
geometric graph whose edges are colored with q colors, yields a monochromatic
convex geometric graph on n vertices.
  For fixed l \geq 2 and q \geq 4, our results determine the correct
exponential tower growth rate for N as a function of n, similar to the usual
hypergraph Ramsey problem, even though we require our monochromatic set to be
in convex position. Our results also apply to the case of l=3 and q=2 by using
a geometric variation of the stepping up lemma of Erd\H os and Hajnal. This is
in contrast to the fact that the upper and lower bounds for the usual 3-uniform
hypergraph Ramsey problem for two colors differ by one exponential in the
tower.
