Let $\varphi\,{:}\,G \,{\rightarrow}\,{\rm O}(d)$ be an orthogonal representation of a group $G$. A finite subset $X$ of a Euclidean space ${\bf R}^d$ is said to be subtransitive under$(G, \varphi)$ when it is a subset of an orbit under the action of ${\varphi }(G)$; furthermore, we say that $X$ is finitely subtransitive when it is subtransitive under $(G, \varphi)$ for some finite group $G$. If $X$ is subtransitive under a representation $\varphi\,{:}\,G\,{\rightarrow}\, {\rm O}(d)$ then it is clear that $X$ is spherical; that is, contained in a sphere in the representation space ${\bf R}^d$. In connection with Euclidean Ramsey theory, I.B. Leader [6] has posed the converse question:

Is is true that every finite spherical set is finitely subtransitive?