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Source. The implicit compactness step in Kříž's Theorem 3.2, published p. 903 (publisher PDF). This is a complete relative deduction, not a separately numbered source lemma.

Statement

Let $F$ be a finite configuration with an equivalence relation $E$. Fix a dimension $m$ and an integer $q\ge1$. Suppose every $q$-coloring of $\mathbb R^m$ has an isometrical embedding of $F$ whose colors are constant on each $E$-class. Then some finite $X\subseteq\mathbb R^m$ has the same property for every coloring $X\to[q]$.

Full proof relative to Rado selection

Suppose no finite $X$ works. For each finite $X\subseteq\mathbb R^m$, choose a coloring $c_X:X\to[q]$ with no such embedding. The finite-choice selection principle gives a coloring $c:\mathbb R^m\to[q]$ agreeing, on every finite $Y$, with $c_X$ for some finite $X\supseteq Y$.

The hypothesis gives a copy $\phi(F)$ on which $c$ respects $E$. Apply the selection property to $Y=\phi(F)$. For some finite $X\supseteq\phi(F)$ the colors $c_X$ agree with $c$ at every point of that copy. Thus the same embedding respects $E$ under $c_X$, contradicting its choice. A finite witness exists. $\square$

Only finitely many point-color constraints are inspected on each copy. No requirement is imposed between distinct $E$-classes. The case of the empty configuration is immediate with $X=\varnothing$.

Related argument. Moore's finite-witness lemma is the universal-relation specialization. The proof above handles the more general equivalence-color constraint needed by Kříž.

Bears on. Problem 174.