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 ${\alpha }_1,\dots ,{\alpha }_r$ 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.