On perfect monochromatic sets in analytic spaces

It is known that if C is a G-delta symmetric relation in an analytic space and there exists an uncountable C-monochromatic set then there exists also a perfect one. We extend this result to the case of countably many G-delta relations, in the same spirit as Mycielski's theorem.
As an application, we obtain the following: If G is a Polish group containing a nonempty dense-in-itslef set generating a free subgroup, then G contains a free subgroup generated by a perfect set.