On monochromatic solutions of equations in groups

We show that the number of monochromatic solutions of the equation x α1 1 x α2 2 ··· x αr r = g in a 2-coloring of a finite group G ,w here α1 ,...,α r are permutations and g ∈ G, depends only on the cardinalities of the chromatic classes but not on their distribution. We give some applications to arithmetic Ramsey statements.