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Source. Published pp. 349–350, Theorem 13 (published scan).

Statement. Theorem 13 (p. 349): “If $K$ is not spherical, then $K$ is not Ramsey.” Here a configuration is spherical when it lies on the surface of a sphere (pp. 348–349), and Ramsey when for each $r$ some $\mathbb R^n$ has a monochromatic congruent copy of $K$ in every $r$-coloring (p. 344). The proof (pp. 349–350) gives more for a finite nonspherical $K$: there is a positive integer $r$ depending only on $K$ such that every $\mathbb R^N$ has an $r$-coloring, constant on spheres about the origin, with no monochromatic congruent copy of $K$. An infinite $K$ is reduced to a finite subset (p. 350).

Complete proof. Write $K=\{v_0,\ldots,v_k\}$. By lemma_14, choose $c_i$ with

$$\sum_i c_i(v_i-v_0)=0,\qquad \sum_i c_i(\|v_i\|^2-\|v_0\|^2)=b\ne0.$$

By lemma_15, there is a finite coloring $\chi$ of $\mathbb R$ with no monochromatic solution of $\sum_i c_i(t_i-t_0)=b$. In every ambient dimension use the same radial rule

$$\chi_N(x)=\chi(\|x\|^2).$$

The vector relation and the nonzero scalar $b$ are unchanged under congruence: orthogonal maps preserve norms; translation by $z$ adds $2\langle z,\sum_i c_i(v_i-v_0)\rangle=0$ to the scalar expression. The Gram extension proves the same conclusion when the copy is in another ambient dimension. If $v_0',\ldots,v_k'$ were a monochromatic copy under $\chi_N$, then the numbers $t_i=\|v_i'\|^2$ would all have the same $\chi$-color and satisfy the forbidden equation. This is impossible. The number of colors came from $K$ alone and is independent of $N$. $\square$

Source precision. Near the end of the proof on p. 350, the source writes squared-norm differences as the colored scalar variables. Translation of scalar arguments need not preserve $\chi$-colors. The variables must be the individual squared norms as above; their differences occur only in the equation. This corrects that final substitution without changing the source's shell-coloring method.

For a set $K$ in a fixed finite-dimensional Euclidean space, the infinite version follows from the finite obstruction argument in finite_sphere_obstruction. The sufficiency of sphericality is a separate classification question; this theorem supplies only necessity.

Uses. The exact obstruction is an input to zero_height_obstruction and the Ramsey-base specialization in two_color_observation.

Bears on. #174.