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• W2511892149 abstract "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." @default.
• W2511892149 created "2016-09-16" @default.
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• W2511892149 date "2014-10-01" @default.
• W2511892149 modified "2023-10-10" @default.
• W2511892149 title "A Ramsey-type result for geometricℓ-hypergraphs" @default.
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• W2511892149 doi "https://doi.org/10.1016/j.ejc.2014.04.001" @default.
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