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Source. Sections 1–2, pp. 3–9 of the arXiv version 1012.1350v1 identified in the source digest. These are conjectures as formulated in this source. Their equivalence does not prove them or settle #174.

A nonempty finite Euclidean set is transitive if a finite group of isometries acts transitively on it. Equivalently, use its full symmetry group on its affine span; rotations fixing that span pointwise are irrelevant. It is subtransitive if it is congruent to a subset of a finite transitive set, possibly in a larger dimension. A set $Y$ is $k$-Ramsey for $X$ if every $k$-coloring of $Y$ contains a monochromatic congruent copy of $X$.

• A (p. 3). A finite set is Ramsey if and only if it is subtransitive.
• B (p. 5). For every finite transitive $X$ and every positive integer $k$, there are a positive integer $n$ and a fixed scale $a>0$ such that $aX^n$ is $k$-Ramsey for $X$.
• C (p. 5). For every finite group $G$ and positive integer $k$, there are positive integers $n,d$ such that every coloring $G^n\to[k]$ admits $g=(g_1,\ldots,g_n)$ and $I\subseteq[n]$, $|I|=d$, for which $\{g\times_I h:h\in G\}$ is monochromatic. Here $(g\times_I h)_i=g_ih$ for $i\in I$ and $g_i$ otherwise.

The scale in B and degree $d$ in C are selected before the coloring. Letting them depend on the coloring weakens these assertions. C without a fixed degree follows from Hales–Jewett; that does not prove C.

A template is a nondecreasing word $\tau\in[m]^\ell$ for some $\ell$ (p. 8). Choose pairwise disjoint blocks $I_1,\ldots,I_\ell\subseteq[n]$ and fixed letters outside their union. For every rearrangement $\pi$ of the multiset of letters of $\tau$, put letter $\pi_j$ on $I_j$. The resulting collection of words is a block set with template $\tau$. Its degree is $\sum_j|I_j|$. Empty blocks are allowed in the general definition. A block set is $t$-uniform if all its blocks have size $t$; a positive uniform degree forces $t\ge1$. A block permutation set uses the template $12\cdots m$.

• D (p. 7). For every $m,k\ge1$, some fixed positive $n,d$ ensure a monochromatic degree-$d$ block permutation set in every $k$-coloring of $[m]^n$.
• E (p. 9). For every $m,k\ge1$ and every template $\tau$, some fixed positive $n,d$ ensure a monochromatic degree-$d$ block set with that template in every $k$-coloring of $[m]^n$.
• F (p. 9). The assertion of E holds with a uniform block set.

The source omits the word “monochromatic” from Conjecture D (p. 7) and from its definition of $k$-Ramsey for $X$ (p. 5); without it both are trivial. The statements here supply it, as the source's proofs use it.

The source proves the following cycle of implications:

$$C\Longrightarrow B\Longrightarrow F\Longrightarrow E \Longleftrightarrow D\Longrightarrow C.$$

See Proposition 2.1, Proposition 2.2, template substitution and Proposition 2.4. Thus B–F are equivalent as universally quantified assertions. They imply the sufficient direction of A. The necessary direction of A is a separate conjecture, and no equivalence of A with B–F is established here.

The paper remarks (p. 6) that degree one cannot be required in C, nor that $g$ be constant on $I$. For a nontrivial $G$, one cannot additionally require $g$ to be constant on $I$ of fixed size $d$. Color a word by which of the two length-$d$ halves of the residue classes modulo $2d$ contains its number of identity letters. If $g_i=a$ throughout $I$, choosing $h=a^{-1}$ adds exactly $d$ identity letters there; choosing any other $h$ adds none. The two colors differ. In particular degree one is impossible for this two-color requirement. This obstruction does not apply to the trivial group.

Bears on. Problem 174: Conjecture A is the paper's proposed characterization of the Ramsey sets, a rival to Graham's conjecture that the Ramsey sets are the spherical ones; B--F are equivalent sufficient conditions for its "if" direction. All six are conjectures as posed in this paper.