Simplices and Regular Polygonal Tori in Euclidean Ramsey Theory

Miltiadis Karamanlis

Published 2021-05-17, updated 2021-06-13Version 2

We show that any finite affinely independent set can be isometrically embedded into a regular polygonal torus, that is, a finite product of regular polygons. As a consequence, with a straightforward application of K\v{r}\'{i}\v{z}'s theorem, we get an alternative proof of the fact that all finite affinely independent sets are Ramsey, a result which was originally proved by Frankl and R\"{o}dl.