---

Pages below are those of the arXiv version 1012.1350v1 identified in the source digest.

Theorems used in the proofs

1. Ramsey's theorem (finite form; the paper gives it no citation, its general reference for Ramsey theory being [8], Graham, Rothschild and Spencer, Ramsey Theory, Wiley, 1980): for positive integers $q,K,L$ with $L\ge q$, some $N$ makes every $K$-coloring of the $q$-subsets of $[N]$ constant on the $q$-subsets of some $L$-set. Used for two-letter templates (p. 12) and twice in Theorem 3.1 (pp. 12--14).
2. Van der Waerden's theorem (reference [14], Beweis einer Baudetschen Vermutung, Nieuw Arch. Wisk. 15 (1927), 212--216): every $K$-coloring of a long enough interval has a monochromatic arithmetic progression of length $L$. Used in Theorem 3.1 with $L=r+1$.
3. Orthogonal equivalence of representations: Lemma 4.7.1 of Wolf, Spaces of Constant Curvature, McGraw-Hill, 1967 (reference [15]), cited in the proof of Theorem 4.2 (p. 17); see the countability step.
4. Sard's theorem, through Milnor, Topology from the Differentiable Viewpoint, 1965 (reference [13]), cited at the end of the proof of Lemma 4.1 (p. 16) for the nullity of a lower-dimensional smooth image.

The Hales--Jewett theorem (reference [9]) is mentioned for comparison only (pp. 6--7): without the fixed size $|I|=d$, Conjectures C and D would follow from it. It is not used in a proof.

Earlier results the paper recalls

• Finite Ramsey witnesses by compactness (p. 5).
• Ramsey sets are spherical, products of Ramsey sets are Ramsey and bricks are Ramsey (p. 2).
• Coloring by distance from the origin, the idea behind the paper's four-coloring that avoids copies of $\{0,1,2\}$ (p. 2).
• Non-degenerate simplices are Ramsey and Kříž's soluble-group theorem (p. 2).

The paper also cites Cantwell on the $120$-cell, Johnson on subtransitive sets, and Kříž's trapezoid paper (p. 18) as background; none of them enters a proof.

Bears on. Problem 174: background to the paper's results, which bear on the problem as their own pages state.