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- W2950377626 abstract "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 ErdH 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 ErdH 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." @default.
- W2950377626 created "2019-06-27" @default.
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- W2950377626 date "2013-05-22" @default.
- W2950377626 modified "2023-09-27" @default.
- W2950377626 title "A Ramsey-type result for geometric l-hypergraphs" @default.
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