Definition ($A$-invariant). For a finite $A\subseteq X$, we say that a colouring of $X^n$ is $A$-invariant if it is invariant under chaning the coordinates within $A$. i.e. for $x=(x_1,\dots ,x_n)$, $x^{\prime }=(x_1^{\prime },\dots ,x_n^{\prime })$ if $\forall i$ either $x_i=x_i^{\prime }$ or $x_i,x_i^{\prime }\in A$ implies $c(x^{\prime })=c(x)$.