A countable metric space H with D as set of distances is universal if it is homogeneous and every finite metric space with set of distances a subset of D has an isometric embedding into H. Two recent results will be discussed:

1. Every countable universal metric space with a finite set of distances is indivisible.

2. Given a countable universal metric space H with D as set of distances, D assumed to be bounded, and given an ǫ > 0 there exists a countable metric subspace K with F as set of distances so that:

(a) F is a finite subset of D.

(b) For every point a in H there is a point a ′ in K so that dH(a,a′ ) < ǫ.

Those results are steps towards the oscillation stability problem for universal metric spaces. The notion of oscillation stability being a very general one coined by Pestov for topological groups which due to a result of Pestov in the case of a metric space H and the group of isometries of a metric space H turns into the problem of deciding whether H is approximately indivisible. That is, is it the case that for every partition of H into finitely many parts and every ǫ > 0 there is an isometric copy of H in H whose distance from one of the parts is at most ǫ. It may appear to be the case, but unfortunately, as already observed by Lopez-Abad and Nguyen Van Th´e, it is not the case that results 1. and 2. imply that every countable universal metric space is approximately indivisible. They showed 1. and 2. for the Urysohn sphere with F the points of an equal partition of the interval [0, 1] into m parts. Then they provided a quite complicated and ingenious construction to show that, given that universal metric spaces whose sets of distances are an initial interval of ω are indivisible, the Urysohn sphere is approximately indivisible. It has been shown by Nguyen Van Th´e and Sauer that universal metric spaces whose sets of distances are an initial interval of ω are indivisible. A direct modification of the Abad, Nguyen construction, using circular paths, will not work in the general case because the infimum of the path length they use may fall into a ”hole” of the set of distances D of the space H. Nevertheless a careful analysis of their approach leads to a modification which for some interesting families of countable universal metric spaces, leads together with results 1. and 2., to the conclusion that those spaces are approximately indivisible.

J. Lopez-Abad and L. Nguyen Van Th´e, The oscillation stability problem for the Urysohn sphere: A combinatorial approach, Topology Appl. 155 Issue 14 (2008) 1516-1530.

L. Nguyen Van Th´e, N. Sauer, The Urysohn Sphere is oscillation stable, Goemetric

and

Functional Analysis 19 Issue 2 (2009) 536-557.

N. Sauer, Vertex partitions of metric spaces with finite distance sets, Discrete Mathematics submitted.

N. Sauer, Approximating universal metric spaces, manuscript.