 A-invariant coloring of a Cartesian power

For $A\subseteq X$, a coloring of $X^n$ is $A$-invariant when changing any coordinates from one member of $A$ to another does not change the color. A product argument upgrades the existence of a copy of $X$ on which $A$ is monochromatic to an $A$-invariant copy of every finite Cartesian power $X^n$.