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Source. Moore, arXiv:2608.09649v1, p. 2, Lemma 2.3 (canonical PDF). Moore states this as a hypergraph compactness consequence, citing de Bruijn–Erdős and Proposition 4 of Euclidean Ramsey theorems I, printed p. 343, PDF p. 3. The argument below explicitly supplies the finite-configuration application.

Statement. Let $T$ be a finite Euclidean configuration and let $n\ge0$ and $r\ge1$ be integers. Write $S\to_r T$ when every $r$-coloring of $S$ has a monochromatic congruent copy of $T$. If $\mathbb R^n\to_r T$, then some finite $A\subseteq\mathbb R^n$ satisfies $A\to_r T$.

External input. Use the Rado selection principle quoted as Theorem 2 on p. 371 of de Bruijn–Erdős (1951). In the constant finite-choice-set case needed here, if every finite $F\subseteq I$ is assigned a function $c_F:F\to\{1,\ldots,r\}$, there is $c:I\to\{1,\ldots,r\}$ such that for every finite $K\subseteq I$ there is a finite $F\supseteq K$ with $c|_K=c_F|_K$. The selection theorem remains external to Moore's paper. Its complete original proof is compiled at Rado (1949), Lemma 1, printed pp. 337–339.

Complete relative proof. The empty target, if allowed, is immediate, so suppose $T\ne\varnothing$. Assume that no finite witness exists. For each finite $F\subseteq\mathbb R^n$, choose an $r$-coloring $c_F$ having no monochromatic copy of $T$. Apply the stated selection principle with $I=\mathbb R^n$ and the finite choice set $\{1,\ldots,r\}$ at every point. It supplies a global coloring $c$.

The hypothesis $\mathbb R^n\to_r T$ gives a finite monochromatic set $K\subseteq\mathbb R^n$ congruent to $T$ under $c$. By the selection property, some finite $F\supseteq K$ satisfies $c_F|_K=c|_K$. Thus $K$ is a monochromatic copy of $T$ in $c_F$, contrary to the choice of $c_F$. A finite witness therefore exists. $\square$

Related proof. Paper I's finite-witness reconstruction gives the same reduction through compactness of a finite-palette product. The argument above uses Rado's selection principle, from which de Bruijn and Erdős deduce their coloring theorem; Moore cites that theorem's hypergraph form.

Scope. This proves the required hypergraph application directly; it does not infer hypergraph compactness merely from the graph coloring statement. It expands Moore's stated lemma without claiming to reprove Rado's theorem.

Bears on. #174.