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Source. Published pp. 341–344, 347, 349, 357 and 360 (published scan).

A configuration is a nonempty finite subset of a Euclidean space unless an infinite-set extension is stated explicitly. Write $R(K,N,r)$ when every coloring $\mathbb R^N\to\{1,\ldots,r\}$ contains a monochromatic congruent copy of $K$. A configuration is Ramsey if for every positive integer $r$ this holds for some $N$. A congruent copy preserves all pairwise distances; its orientation is unrestricted, but its scale is fixed.

The source's $\ell$-Ramsey convention means that for every number $r$ of available colors, some dimension forces a copy using at most $\ell$ colors. Thus $1$-Ramsey is Ramsey. This differs from the fixed-color sense of “$r$-Ramsey” in the abstract on p. 341, where a relation $R\subseteq A\times B$ is $r$-Ramsey when every partition of $B$ into $r$ parts puts the set $R(a)=\{b:(a,b)\in R\}$ of some $a\in A$ inside one part; some later papers use the phrase in that fixed-color sense too.

A set is spherical if it lies on one sphere, possibly in a larger ambient space. Its intrinsic circumradius is the radius about the unique circumcenter in its affine hull. A singleton has intrinsic radius zero. Sphere-Ramsey here means that for every $r$ there are $N_0,R_0$ such that all spheres of radius $R\ge R_0$ in $\mathbb R^N$, $N\ge N_0$, force a monochromatic copy. We count ambient dimension; the sphere itself has dimension $N-1$. The definition on p. 360 asks for “a sphere $S$ of dimension at least $n$ and radius at least $d$” without saying which dimension is meant; since $n$ is existentially quantified, the two readings define the same property. This property does not require radii arbitrarily close to the intrinsic radius.

Complete proof of elementary facts. Subsets inherit every forcing statement by restricting the forced copy. Applying an inverse similarity to a coloring proves that similarities preserve the Ramsey property and all fixed-dimension color bounds. Larger ambient dimensions preserve a bound by restriction to a suitable subspace. Singleton statements are immediate.

For a regular simplex on $k+1$ vertices with edge length $a>0$, start with $(a/\sqrt2)e_i$ in $\mathbb R^{k+1}$ and subtract their mean. The affine hull has dimension $k$, every pair distance is $a$, and the common squared norm is $a^2k/[2(k+1)]$. A regular simplex on $kr+1$ vertices has $k+1$ vertices of one color under any $r$-coloring, by the pigeonhole principle. Thus the original simplex satisfies $R(K,kr,r)$ for $k\ge1$.

A distance-preserving correspondence between two finite configurations preserves the Gram matrix of differences from any chosen base point, by $2\langle u,v\rangle=\|u\|^2+\|v\|^2-\|u-v\|^2$. It therefore extends to a linear isometry between their difference spans: any relation has squared norm zero on one side exactly when it does on the other. This proves that affine relations and affine dimension are preserved by congruence, including when the configurations are placed in different ambient dimensions.

For a spherical configuration, orthogonally project any center onto its affine hull. Pythagoras shows that the projected point is still equidistant from all configuration points, with the squared radius decreased by the same nonnegative constant. Two such centers in the affine hull have a difference orthogonal to every difference of configuration points, hence to the hull's direction space; that difference must be zero. This proves existence and uniqueness of the intrinsic circumcenter. $\square$

Bears on. #174.