Showing papers by "Norbert Sauer published in 2007"
TL;DR: In this article, it was shown 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.
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.
58 citations
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TL;DR: Finite dimensional partition properties of the countable homogeneous dense local order (a directed graph closely related to the order structure of the rationals) are studied.
Abstract: We study the finite dimensional partition properties of the countable homogeneous dense local order. Some of our results use ideas borrowed from the partition calculus of the rationals and are obtained thanks to a strengthening of Milliken's theorem on trees.
19 citations
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TL;DR: In this article, it was shown that a countable ultrametric space embeds isometrically into an indivisible metric space if and only if it does not contain a strictly increasing sequence of balls.
Abstract: A metric space is indivisible if for any partition of it into finitely many pieces one piece contains an isometric copy of the whole space. Continuing our investigation of indivisible metric spaces, we show that a countable ultrametric space embeds isometrically into an indivisible ultrametric metric space if and only if it does not contain a strictly increasing sequence of balls.
17 citations
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TL;DR: In this article, the oscillation stability problem for the Urysohn sphere was solved by solving a purely combinatorial problem involving a family of countable homogeneous metric spaces with finitely many distances.
Abstract: We solve the oscillation stability problem for the Urysohn sphere, an analog of the distortion problem for the Hilbert space in the context of the Urysohn universal metric space. This is achieved by solving a purely combinatorial problem involving a family of countable homogeneous metric spaces with finitely many distances.
3 citations