A Ramsey-type result for geometric l-hypergraphs

TL;DR

This paper establishes exponential bounds for a geometric Ramsey problem involving convex configurations in the plane, combining hypergraph Ramsey theory with the Erdős–Szekeres theorem, and extends known results to new parameter ranges.

Contribution

It introduces new exponential bounds for geometric Ramsey numbers involving convex sets in the plane, bridging hypergraph Ramsey theory and geometric combinatorics.

Findings

For l=2, established a single exponential bound on N.

For fixed l≥2 and q≥4, determined the exponential tower growth rate of N.

Applied a geometric variation of the stepping-up lemma for l=3, q=2.

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 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.