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Source. Published pp. 360–362, Theorem 25 (published scan).

Statement. Let $\ell\ge2$. If $K$ cannot be contained in at most $\ell-1$ concentric spheres, then it is not $m$-Ramsey for every integer $1\le m<\ell$, in the source's few-color convention. There is one finite color count, independent of ambient dimension, for which every congruent copy uses at least $\ell$ colors.

Complete proof. First assume $K$ finite. Consider each partition $P$ of $K$ into at most $\ell-1$ nonempty classes. There are finitely many such partitions. In each class choose an anchor and form its pairs with every other point in that class. A point equidistant across all these pairs would be the common center of at most $\ell-1$ spheres covering $K$, contrary to the hypothesis. Therefore lemma_27 supplies a finite radial coloring $\chi_P$ which prevents any congruent copy from making all classes of this particular partition monochromatic.

Take the product of these finitely many colorings. Its number of colors depends only on $K$ and $\ell$, not on the ambient dimension. If a copy used at most $\ell-1$ product colors, pull its color classes back to a partition $P$ of $K$. Every such class would also be monochromatic for the coordinate $\chi_P$, contradicting that coordinate's defining property. Hence every copy uses at least $\ell$ colors.

For $K$ in any fixed finite-dimensional Euclidean space, possibly infinite, finite_sphere_obstruction gives a finite subset $S$ that still does not fit in $\ell-1$ concentric spheres. Apply the finite result to $S$. Every congruent copy of $K$ contains a congruent copy of $S$, so the same coloring works for $K$. $\square$

Endpoint. The source states $m<\ell$, not $m\le\ell$. In particular the $\ell=2$ case gives the necessary sphericity condition for $1$-Ramsey sets; it does not say that a nonspherical set cannot be $2$-Ramsey.

Bears on. #174.