Abstract: We consider polarized partition relations concerning partitions into an infinite number of pieces and also partitions defined on products of higher dimension. We use an infinite version of the method of induced coloring which is frequent in Finite Ramsey Theory. Sufficient conditions on cardinals, 2,. ,, B are given in order to satisfy the polarized partition relation ? 22 1,1,...,1 It is shown that the simplest infinite dimensional polarized partition relations fail under the assumption of the axiom of choice, and that under certain large cardinal hypotesis, there are valid polarized partition relations defined on the union of all the finite dimensional powers of a cardinal Ver menos