Simplices and Regular Polygonal Tori in Euclidean Ramsey Theory

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