On monochromatic solutions of equations in groups

Abstract

We show that the number of monochromatic solutions of the equation $x_1^{\alpha_1}x_2^{\alpha_2}\cdots x_r^{\alpha_r}=g$ in a $2$-coloring of a finite group $G$, where $\alpha_1,\ldots,\alpha_r$ are permutations and $g\in G$, depends only on the cardinalities of the chromatic classes but not on their distribution. We give some applications to arithmetic Ramsey statements.