Euclidean Ramsey set

A finite set $X\subseteq {\mathbb{R}}^d$ is Euclidean Ramsey when, for every number of colors, some finite-dimensional Euclidean space has the property that every coloring of it contains a monochromatic isometric copy of $X$.

A finite point set is spherical when it lies on a sphere. Every Euclidean Ramsey set is spherical. Whether every finite spherical point set is Euclidean Ramsey is open.

If finite point sets $X$ and $Y$ are Euclidean Ramsey, then their orthogonal Cartesian product $X\times Y$ is Euclidean Ramsey. The proof first chooses a finite Ramsey witness for $X$, colors a witness for $Y$ by the complete color pattern it induces on the first witness, and then applies the two Ramsey properties in succession.

Every nondegenerate triangle is a Euclidean Ramsey set.

Every line segment is a Euclidean Ramsey set.

Every regular polygon is a Euclidean Ramsey set. A Hales-Jewett theorem combinatorial line in a Cartesian power of its vertex set is a scaled regular polygon; including finitely many reciprocal square-root scalings makes one such line isometric to the original polygon.

A finite Euclidean configuration is edge Ramsey when every finite edge coloring of a suitable finite Euclidean set contains a monochromatic isometric copy. The edge Ramsey configurations are exactly the equidistant sets, equivalently the vertex sets of regular simplices.

A finite point set is cyclic transitive when a cyclic group of isometries acts transitively on it.

Every finite cyclic transitive point set is a Euclidean Ramsey set. In particular every regular polygon is Euclidean Ramsey.