Divisibility of countable metric spaces

Abstract

Prompted by a recent question of Hjorth [G. Hjorth, An oscillation theorem for groups of isometries, manuscript] 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 bounded and totally Cantor disconnected, which implies in particular that every Urysohn space U V with V containing some dense 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.