A Ramsey-type result for geometric [ell][ell]-hypergraphs

Let n greater than or equal to [ell] greater than or equal to 2n greater than or equal to [ell] greater than or equal to 2 and q greater than or equal to 2q greater than or equal to 2. We consider the minimum NN such that whenever we have NN points in the plane in general position and the [ell][ell]-subsets of these points are colored with qq colors, there is a subset SS of nn points all of whose [ell][ell]-subsets have the same color and furthermore SS is in convex position. This combines two classical areas of intense study over the last 75 years: the Ramsey problem for hypergraphs and the Erdos-Szekeres theorem on convex configurations in the plane. For the special case [ell]=2[ell]=2, we establish a single exponential bound on the minimum NN, such that every complete NN-vertex geometric graph whose edges are colored with qq colors, yields a monochromatic convex geometric graph on nn vertices. For fixed [ell] greater than or equal to 2[ell] greater than or equal to 2 and q greater than or equal to 4q greater than or equal to 4, our results determine the correct exponential tower growth rate for NN as a function of nn, 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 [ell]=3[ell]=3 and q=2q=2 by using a geometric variation of the stepping up lemma of Erdos 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.