Vertex partitions of metric spaces with finite distance sets

A metric space M=(M,d) is indivisible if for every colouring χ:M→2 there exists i∈2 and a copy N=(N,d) of M in M so that χ(x)=i for all x∈N. The metric space M is homogeneous if for every isometry α of a finite subspace of M to a subspace of M there exists an isometry of M onto M extending α. A homogeneous metric space UD with D as set of distances is an Urysohn metric space if every finite metric space with set of distances a subset of D has an isometry into UD. The main result of this paper states that all countable Urysohn metric spaces with a finite set of distances are indivisible.

► Which metric spaces are oscillation stable? ► A major advance in answering this question is the following theorem, which is proved in this paper: every homogeneous metric space with finite distance set is indivisible.