Coding power of product of partitions

Given two combinatorial notions $\mathsf{P}_0$ and $\mathsf{P}_1$, can we encode $\mathsf{P}_0$ via $\mathsf{P}_1$. In this talk we address the question where $\mathsf{P}_0$ is 3-coloring of integers and $\mathsf{P}_1$ is product of finitely many 2-colorings of integers. We firstly reduce the question to a lemma which asserts that certain $\Pi^0_1$ class of colorings admit two members violating a particular combinatorial constraint. Then we took a digression to see how complex does the class has to be so as to maintain the cross constraint. We weaken the two members in the lemma in certain way to address an open 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 strengthen of the lemma is a basis theorem for $\Pi^0_1$ class with additional constraint. We look at several such variants of basis theorem, among them some are unknown. We end up by introducing some results and questions concerning product of infinitely many colorings.