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Source. Published p. 360, Theorem 24 (published scan).

Statement. Every finite brick, and therefore every nonempty subset of its vertices, is sphere-Ramsey in the large-radius convention of definitions. More generally the orthogonal product of two finite sphere-Ramsey configurations is sphere-Ramsey.

Complete proof. A regular simplex on $r+1$ vertices with edge length $a>0$ has circumradius $\rho=a\sqrt{r/[2(r+1)]}$ and lies in an $r$-dimensional linear space about its center. For every $R\ge\rho$, add the common orthogonal coordinate $\sqrt{R^2-\rho^2}$; the vertices then lie on the sphere of radius $R$ in $\mathbb R^{r+1}$. Any $r$-coloring of this sphere has two of the vertices of the same color, at distance $a$. Thus two-point configurations are sphere-Ramsey.

Let $K_1,K_2$ be sphere-Ramsey and fix $r$. Choose a sphere in $\mathbb R^{N_1}$ of radius $R_1$ forcing $K_1$ in $r$ colors. The finite-witness compactness argument gives a finite subset $T$ of that sphere still forcing $K_1$; put $t=|T|$. Then choose a sphere in $\mathbb R^{N_2}$ of radius $R_2$ forcing $K_2$ in $r^t$ colors, and a finite witness $S$ on it. Translate the centers to zero before taking the orthogonal product. Every point of $T\times S$ has norm $R_*:=\sqrt{R_1^2+R_2^2}$.

The pattern-color proof of theorem_20 shows that every $r$-coloring of this product contains a monochromatic copy of $K_1\times K_2$. For every $R\ge R_*$, send each point $z$ of the finite product to $(z,\sqrt{R^2-R_*^2})$. This preserves all distances and places the witness on the radius-$R$ sphere in $\mathbb R^{N_1+N_2+1}$. Zero-padding gives the same witness in every larger ambient dimension. Arbitrary sphere colorings restrict to it, proving the required uniform large-radius and large-dimension statement.

Iterate this product result over the two-point factors of a brick, and restrict a forced brick copy to the desired subset. Degenerate factors are deleted and singletons are immediate. $\square$

The source's references to “Theorem 14” in the product discussion should refer to Theorem 20. The explicit extra coordinate above supplies the all-larger-radii clause of its definition. This conclusion does not claim witnesses on every sphere of radius just above a subset's own intrinsic circumradius; that is the stronger hyper-Ramsey issue distinguished in corollary_6_5.

Bears on. #174.