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A finite configuration is a nonempty finite subset of a Euclidean space. It is Ramsey if, for every integer $k\ge1$, some integer $N\ge1$ has the property that every $k$-coloring of $\mathbb R^N$ contains a monochromatic congruent copy of the configuration. There is no measurability requirement.

A configuration is transitive if a finite group of ambient isometries preserves it and acts transitively on its points. It is soluble if such a group can be chosen soluble. It is subtransitive, respectively subsoluble, if it is congruent to a subset of a finite transitive, respectively soluble, configuration in some Euclidean space. The enclosing space may have greater dimension. Solubility refers to a chosen transitive subgroup; the full symmetry group need not be soluble.

For a group $H$, write $H^{(0)}=H$ and let $H^{(i+1)}$ be the subgroup generated by commutators of elements of $H^{(i)}$. The group is soluble when $H^{(r)}=1$ for some nonnegative integer $r$.

Elementary facts. Finite isometry groups have a fixed point. Every finite transitive configuration is spherical. Subsets and congruent copies preserve the subtransitive, subsoluble and Ramsey properties. If $H$ is soluble, then so are its subgroups, homomorphic images, finite direct powers, and an extension of $H$ by a soluble group. In particular, $H^{n+1}\rtimes C_{n+1}$ is soluble. Cartesian products of finite transitive configurations are transitive under the product of their chosen groups.

Complete proof. If a finite isometry group $G$ acts on $\mathbb R^d$, average an arbitrary orbit:

$$v=\frac1{|G|}\sum_{g\in G}g(p).$$

Each isometry is affine, so $hv=v$ for every $h\in G$. Translation by $-v$ turns the action into an orthogonal linear action. Every orbit then has a constant norm, proving sphericity, with radius zero allowed for a singleton. A congruent copy or a subset of a subset of an enclosing transitive set has the same type of enclosure. A monochromatic copy of an enclosing Ramsey set contains a monochromatic copy of every specified subset. Congruence transports that assertion without changing distances.

Derived subgroups of a subgroup are contained in the corresponding derived subgroups of its ambient group. Homomorphisms carry commutators to commutators. Commutators in a direct product are computed coordinate by coordinate; hence a finite product of soluble groups is soluble. If $N\mathrel{\triangleleft}K$, $N^{(s)}=1$, and $(K/N)^{(r)}=1$, then $K^{(r)}\subseteq N$ and $K^{(r+s)}=1$. This proves the extension assertion. Cyclic groups are abelian, so the assertion applies to $H^{n+1}\rtimes C_{n+1}$.

Finally, coordinatewise isometries preserve the sum of squared coordinate distances. Given two elements of a product of transitive configurations, choose an isometry taking each coordinate of the first to the corresponding coordinate of the second. Their product acts transitively. If all chosen groups are soluble, their product is soluble. $\square$

These are complete elementary expansions of the conventions and group facts used on source pp. 1–6, arXiv:2606.13472v1. The deep assertion that every subsoluble set is Ramsey is the exact external Kříž input, not a consequence of these elementary facts alone.

Bears on. #174.