Simplices and Regular Polygonal Tori in Euclidean Ramsey Theory

TL;DR

This paper demonstrates that any finite affinely independent set can be embedded into a regular polygonal torus, providing an alternative proof that all such sets are Ramsey, thus contributing to Euclidean Ramsey theory.

Contribution

It introduces a new embedding technique into regular polygonal tori and offers an alternative proof for the Ramsey property of finite affinely independent sets.

Findings

Finite affinely independent sets can be embedded into regular polygonal tori.

An alternative proof that all finite affinely independent sets are Ramsey.

The embedding technique simplifies existing proofs in Euclidean Ramsey theory.

Abstract

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.