Euclidean ramsey theorems. I

Abstract

The general Ramsey problem can be described as follows: Let A and B be two sets, and R a subset of A × B. For a ϵ A denote by R(a) the set {b ϵ B | (a, b) ϵ R}. R is called r-Ramsey if for any r-part partition of B there is some a ϵ A with R(a) in one part. We investigate questions of whether or not certain R are r-Ramsey where B is a Euclidean space and R is defined geometrically.