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Source. Moore, arXiv:2608.09649v1, p. 2, Theorem 2.1 (canonical PDF). The original result is Theorem 20 of Erdős, Graham, Montgomery, Rothschild, Spencer and Straus, Euclidean Ramsey theorems. I (1973), printed p. 357, PDF p. 17; see the original source.

Statement. A finite Euclidean set $T$ is Ramsey when, for every positive integer $r$, some dimension $N$ has the property that every map $\mathbb R^N\to\{1,\ldots,r\}$ gives a monochromatic set congruent to $T$. If the finite sets $A\subseteq\mathbb R^a$ and $B\subseteq\mathbb R^b$ are Ramsey, then

$$A\times B=\{(x,y):x\in A,\ y\in B\}\subseteq\mathbb R^{a+b}$$

is Ramsey, with the usual Euclidean product metric. Repetition gives the same assertion for any finite number of factors.

Proof scope. This is an exact input from the earlier paper. Its complete canonical reconstruction is Theorem 20, with the finite compactness principle. The proof is linked rather than duplicated on this page. Moore applies the theorem to a Ramsey base and an affinely independent auxiliary set in Theorem 1.2.

Bears on. #174.