Partitions of metric spaces with finite distance sets

Abstract: A metric space $\mathrm{M}=(M,\de)$ is {\em indivisible} if for every colouring $\chi: M\to 2$ there exists $i\in 2$ and a copy $\mathrm{N}=(N, \de)$ of $\mathrm{M}$ in $\mathrm{M}$ so that $\chi(x)=i$ for all $x\in N$. The metric space $\mathrm{M}$ is {\em homogeneus} if for every isometry $\alpha$ of a finite subspace of $\mathrm{M}$ to a subspace of $\mathrm{M}$ there exists an isometry of $\mathrm{M}$ onto $\mathrm{M}$ extending $\alpha$. A homogeneous metric space $\mathrm{U}$ with set of distances $\mathcal{D}$ is an Urysohn metric space if every finite metric space with set of distances a subset of $\mathcal{D}$ has an isometry into $\mathrm{U}$. The main result of this paper states that all countable Urysohn metric spaces with a finite set of distances are indivisible.