The coding power of a product of partitions

Given two combinatorial notions P-0 and P-1, can we encode P-0 via P-1 In this paper we address the question where P0 is 3-coloring of integers and P1 is a product of finitely many 2-colorings of integers. We firstly reduce the question to a lemma which asserts that certain.01-classes of colorings admit two members violating a particular combinatorial constraint. Then we digress to see how complex the class must be to maintain the constraint. We weaken the two members in the lemma in a certain way to address a question of Cholak, Dzhafarov, Hirschfeldt and Patey, concerning a sort of Weihrauch degree of stable Ramsey's theorem for pairs. It turns out the resulted strengthening of the lemma is a basis theorem for a.Pi(0)(1)-class with additional constraints. We look at several such variants of basis theorem, among ahich some are unknown. We end up by introducing some results and questions concerning a product of infinitely many colorings.