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

Abstract

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