  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.
