Divisibility of countable metric spaces

Abstract

Prompted by a recent question of G. Hjorth [12] as to whether a bounded Urysohn space is indivisible, that is to say has the property that any partition into finitely many pieces has one piece which contains an isometric copy of the space, we answer this question and more generally investigate partitions of countable metric spaces.We show that an indivisible metric space must be totally Cantor disconnected, which implies in particular that every Urysohn space U V with V bounded or not but dense in some initial segment of R + , is divisible. On the other hand we also show that one can remove "large" pieces from a bounded Urysohn space with the remainder still inducing a copy of this space, providing a certain "measure" of the indivisibility. Associated with every totally Cantor disconnected space is an ultrametric space, and we go on to characterize the countable ultrametric spaces which are homogeneous and indivisible.

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

“…Remark that under our assumptions on S, the space U S always exists and is unique, according to Theorem 1.4 and Example 1.5.3 in [6]. Now in the case S = {0, 1, 2} (a convex subset of the semigroup N of natural numbers) one recovers Hrushovski's theorem 3.7, while the case S = R gives the Solecki-Vershik theorem 3.8.…”

A finite presentation of the rational Urysohn space

“…U X need not exist as is demonstrated by the failure of the amalgamation property. However this is characterized in [4] by 4-value property.…”

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

“…The following case to consider is S 3 , which turns out to be another particular case thanks to an observation made in [3].…”

“…We do not write more here but the interested reader is referred to [3], section on the indivisibility of Urysohn spaces, for more details.…”