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 U D 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 U D . The main result of this paper states that all countable Urysohn metric spaces with a finite set of distances are indivisible.