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arXiv · 1511.09322

A rigid Urysohn-like metric space

Jan Grebík
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Abstract

Recall that the Rado graph is the unique countable graph that realizes all one-point extensions of its finite subgraphs. The Rado graph is well-known to be universal and homogeneous in the sense that every isomorphism between finite subgraphs of $R$ extends to an automorphism of $R$. We construct a graph of the smallest uncountable cardinality $ω_1$ which has the same extension property as $R$, yet its group of automorphisms is trvial. We also present a similar, although technically more complicated, construction of a complete metric space of density $ω_1$, having the extension property like the Urysohn space, yet again its group of isometries is trivial. This improves a recent result of Bielas.

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Jan Grebík. 2018-07-15. A rigid Urysohn-like metric space. https://arxiv.org/abs/1511.09322

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