Finite euclidean ramsey theory

Abstract

Let K be a configuration, a set of points in some finite dimensional Euclidean space. Let n and k be positive integers. The notation R(K, n, r) is an abbreviation for the following statement: For every r-coloring of the points of n-dimensional Euclidean space, Rn, there exists a monochromatic configuration L which is congruent to K. In this paper, it is shown that when K is a square of side l, it can be proved that R(K, 4, 2) holds. When K consists of two points at unit distance, it is also proved that R(K, 4, 6) and R(K, 5, 8) hold.