On monochromatic solutions of equations in groups

We show that the number of monochromatic solutions of the equation x 1 α1x2 α2⋯x r αr = g in a 2-coloring of a finite group G, where α1,...,αr 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.