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Statement

Remark (§2, p. 9, unnumbered). Conjectures D and E (see the conjecture page) are equivalent. D is the case of E for the template $12\ldots m$; and E for a template $\tau_1\ldots\tau_\ell$ on $[m]$ follows from D on $[\ell]$.

Proof sketch

P. 9. Given a coloring of $[m]^n$, color $(x_1,\ldots,x_n)\in[\ell]^n$ by the color of $(\tau_{x_1},\ldots,\tau_{x_n})$ and apply D. Under this map the block permutation words become the words of a block set with template $\tau$ on the same blocks, so the degree, and uniformity when present, are kept. The same step reduces Conjecture F to the template $1\ldots m$ in Proposition 2.4 (p. 12).

Source. Imre Leader, Paul A. Russell and Mark Walters, Transitive sets in Euclidean Ramsey theory, J. Combin. Theory Ser. A 119 (2012), no. 2, 382--396, doi:10.1016/j.jcta.2011.09.005; page from the arXiv version 1012.1350v1 identified in the source digest.

Read depth. Claims checked: the remark (p. 9) was read in full and followed.

Bears on

• Problem 174: one equivalence in the paper's chain showing its Conjectures B--F equivalent. It proves no set Ramsey.