Refinement of a finite coloring

A refinement of a finite coloring assigns different new colors whenever the original colors differ. A set monochromatic for the refinement is therefore monochromatic for the original coloring. Given colorings $\chi ,\psi$, the product coloring $n\mapsto (\chi (n),\psi (n))$ refines both. This makes it possible to impose finitely many coloring obstructions simultaneously.