  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.

Logic