The oscillation stability problem for the Urysohn sphere: A combinatorial approach

Abstract

We study the oscillation stability problem for the Urysohn sphere, an analog of the distortion problem for 2 in the context of the Urysohn space U. In particular, we show that this problem reduces to a purely combinatorial problem involving a family of countable ultrahomogeneous metric spaces with finitely many distances.

Oscillation of Urysohn Type Spaces

“…It is then easily seen that the disjoint amalgamation property implies that every Katȇtov functions has infinitely many realizations. This then in turn implies Lemma 4.1 via a standard construction as for the special case proven in [12].…”

“…Subsequently to the Odell-Schlumprecht result, it was therefore natural to ask whether the Urysohn sphere U [0,1] has distortion as well. The first major step in resolving this question is the main result achieved by Lopez-Abad and Nguyen Van Thé in [12]: The Urysohn sphere is oscillation stable if all Urysohn metric spaces U n for n = {0, 1, . .…”

A theorem of Hrushovski–Solecki–Vershik applied to uniform and coarse embeddings of the Urysohn metric space

“…The union of an infinite chain of iterated n-fold Katětov extensions of X can be verified to be isometric to U Z (cf. [26]). If G is a group acting on X by isometries, then the action lifts to the Katětov extension E Z (X) through the left regular representation, and the Kuratowski embedding is G-equivariant under this lifting.…”

“…This observation generalizes as follows: every metric space is coarsely equivalent to its subspace forming an ε-net for some ε > 0. As one of the authors of [26] (Lionel Nguyen Van Thé) has pointed out to the present author, the proof of Proposition 1 (Section 2.1 in [26]) can be modified so as to establish the following result. Since the composition of two coarse embeddings is a coarse embedding, and every coarse equivalence is a coarse embedding, it follows from Corollary 2.4 that for the purpose of considering coarse embeddings, there is no difference between U and U Z .…”

Vertex partitions of metric spaces with finite distance sets

“…This then led to the question whether the other prominent bounded metric space with a large isometry group, namely the Urysohn sphere U ℜ∩[0, 1] , is oscillation stable. After an initial reformulation of the problem by V. Pestov, see [8], Lopez-Abad and Nguyen Van Thé, see [4], started a programme to reduce the problem to one of discrete mathematics. In particular they proved that the Urysohn sphere will be oscillation stable if and only if each of the Urysohn spaces U m is indivisible.…”