Monochromatic Regular Polygons in finitely colored $\mathbb{Z}_2 *\mathbb{Z}_2 *\mathbb{Z}_2$

TL;DR

This paper proves that in any finite coloring of the free product of three cyclic groups of order two, there always exists a monochromatic regular polygon of any size, with bounds on the edge length.

Contribution

It establishes the existence of monochromatic regular polygons in finitely colored free products of three groups, extending geometric Ramsey theory to this algebraic setting.

Findings

Existence of monochromatic regular k-gons for all k

Bounds on the edge length of these polygons

Application of combinatorial and geometric methods

Abstract

We show that for any finite coloring of the group $\mathbb{Z}_2 *\mathbb{Z}_2 *\mathbb{Z}_2$ and for any positive integer $k$, there always exists a monochromatic regular $k$-gon in $\mathbb{Z}_2 *\mathbb{Z}_2 *\mathbb{Z}_2$ with respect to the word length metric induced by the standard generating set; the edge length of which is estimated.