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.

The collinear set $\{0,1,2\}$ is not a Euclidean Ramsey set. In every dimension, coloring $x$ by $\lfloor 2\Vert x{\Vert }^2\rfloor$ modulo ten prevents a monochromatic congruent copy; the parallelogram law would force the corresponding three integer parts to have a second difference strictly between two and six, which cannot be divisible by ten.

Every nondegenerate triangle is a Euclidean Ramsey set.

Every line segment is a Euclidean Ramsey set.

Every regular polygon is a Euclidean Ramsey set. Its cyclic rotation acts transitively on the vertices, so this is a special case of the Kriz theorem for cyclic transitive point sets.

A finite Euclidean set $X$ is approximately Ramsey if, for every number of colors and every $\epsilon >0$, some finite set has a monochromatic subset whose corresponding pairwise distances differ from those of $X$ by less than $\epsilon$.

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.

For $A\subseteq X$, a coloring of $X^n$ is $A$-invariant when changing any coordinates from one member of $A$ to another does not change the color. A product argument upgrades the existence of a copy of $X$ on which $A$ is monochromatic to an $A$-invariant copy of every finite Cartesian power $X^n$.