A Ramsey-Type Result for Geometric ℓ-Hypergraphs

Let n ≥ ℓ ≥ 2 and q ≥ 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 Erdős-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 ≥ 2 and q ≥ 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 Erdős 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