---

Source. Published p. 361, Lemma 27 (published scan).

Statement. Suppose finitely many pairs $x_i,y_i\in\mathbb R^d$ have no common equidistant point. There is a finite number of colors, independent of ambient dimension, and a radial coloring in each $\mathbb R^N$ such that every congruent copy of the labeled configuration has at least one pair with different colors.

Complete proof. By lemma_26, there are $c_i$ such that

$$\sum_i c_i(x_i-y_i)=0,\qquad \sum_i c_i(\|x_i\|^2-\|y_i\|^2)=b\ne0.$$

Apply theorem_16 over $\mathbb R$ to obtain $\chi$ forbidding $\sum_i c_i(u_i-v_i)=b$ whenever $\chi(u_i)=\chi(v_i)$ for all $i$. Color $z\in\mathbb R^N$ by $\chi(\|z\|^2)$.

The displayed vector relation and its scalar $b$ are preserved by translations, orthogonal maps and embeddings of the difference span, exactly as in theorem_13. If a congruent copy had every pair equally colored, its squared norms would solve the forbidden scalar equation. This is impossible. $\square$

The source's reference to “Lemma 20” in this proof is a numbering error; the required common-bisector criterion is Lemma 26.

Bears on. #174.