Proposition 3.7 (Our product argument). Assuming that:

$X \subseteq ℝ^{d}$
$A \subseteq X$
$\forall k, \exists$ a finite $S \subseteq ℝ^{e}$ such that whenever $S$ is $k$ -coloured, there exists a copy of $X$ that is constant on $A$

Then for all

n, k

, there exists

S^{'}

finite such that whenever

S^{'}

k

-coloured there exists a copy of

X^{n}

that is

A

-invariant.