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The following statements are imported. Their original statements and indicated definitions were checked; their full proofs are outside this source unit.

Kříž's soluble-group theorem. If a finite configuration $F$ admits a soluble group of isometries acting transitively on $F$, then $F$ is Ramsey. Every configuration congruent to a subset of such an $F$ is therefore Ramsey. This is the transitive specialization of Theorem 4.3 in Igor Kříž, Permutation groups in Euclidean Ramsey Theory, Proc. Amer. Math. Soc. 112 (1991), 899–907, DOI 10.1090/S0002-9939-1991-1065087-9. The printed statement on p. 906 (PDF p. 8) says that $F$ is $E_G$-Ramsey, where $E_G$ is orbit equivalence. Transitivity makes this the universal relation, so the conclusion is an ordinary monochromatic copy. The subset step is proved in the elementary closure proof. No assertion here identifies the full symmetry group with the chosen soluble subgroup.

Spherical necessity. Every finite Euclidean Ramsey set is spherical. Equivalently, a nonspherical finite set is not Ramsey. This is Theorem 13, printed p. 349 (PDF p. 9), in Erdős, Graham, Montgomery, Rothschild, Spencer and Straus, Euclidean Ramsey Theorems I (1973). The theorem's proof, including its coloring lemma, is not reproduced here. A circumcenter can be chosen in the affine hull by orthogonally projecting any circumcenter to that hull; all squared radii lose the same squared perpendicular distance.

Uniform block family. For every pair of positive integers $r,s$ and every integer $k\ge1$, there are positive integers $N,d$ such that every $k$-coloring of $[3]^N$ has a monochromatic uniform block set of template $1^r2^s3$, with every block of size $d$. This is Theorem 3.1 together with the uniformity of its construction in Leader, Russell and Walters, Transitive sets in Euclidean Ramsey theory. The inspected author manuscript is dated 22 November 2010: the theorem is on p. 12, the construction on p. 14 has $|I_j|=t$ for every $j$, and p. 15 explicitly records uniformity. The later publication is J. Combin. Theory Ser. A 119 (2012), 382–396, DOI 10.1016/j.jcta.2011.09.005. No byte or full-text equivalence with the published version is asserted. That manuscript calls the total active size $D=\sum_j|I_j|$ the degree; for this family $D=(r+s+1)d$. Ivan–Leader–Walters instead call the common block size $d$ the degree. The displayed input uses the latter convention.

The full uniform block-sets conjecture is equivalent to the assertion that, for every finite transitive $X$ and every $k$, there are $n$ and $s>0$ for which every $k$-coloring of $X^n$ contains a monochromatic congruent copy of $sX$. This is an external equivalence from Section 2 of that same author manuscript (Conjectures B–F and Propositions 2.1, 2.2 and 2.4), not a proof of either conjecture. Its general proof, and the stronger finite-power formulation of Kříž's machinery discussed on ILW p. 7, are not reconstructed here. The particular block conclusions accompanying ILW Theorem 7 use the precise uniform family above, as shown in the complete relative deduction.

The source snapshot identifies the inspected external PDFs and the selected-page reading scope. None is a claim of a new local formal verification or of an included proof of the deep external machinery.

Bears on. #174.