Simplices and Regular Polygonal Tori in Euclidean Ramsey Theory

Abstract

We show that any finite affinely independent set can be isometrically embedded into a regular polygonal torus, that is, the finite product of the vertex sets of some regular polygons. We apply this result in the context of Euclidean Ramsey Theory, highlighting the connection between the two most significant results in this field which were previously thought to be independent. In particular with a straightforward application of Kříž’s theorem we give an alternative proof of the fact that all finite affinely independent sets are Ramsey, a result which was originally proved by Frankl and Rödl.