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A rigid Urysohn-like metric space

Author: Jan Grebík
Journal: Proc. Amer. Math. Soc. 145 (2017), 4049-4060
MSC (2010): Primary 03C50, 05C63
Published electronically: March 23, 2017
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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 $ \omega _1$ which has the same extension property as $ R$, yet its group of automorphisms is trivial. We also present a similar, although technically more complicated, construction of a complete metric space of density $ \omega _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
Affiliation: Institute of Mathematics, Czech Academy of Sciences, 115 67 Prague, Czech Republic

Keywords: Amalgamation, Rado graph, Urysohn space
Received by editor(s): December 7, 2015
Received by editor(s) in revised form: September 19, 2016, September 21, 2016, and September 29, 2016
Published electronically: March 23, 2017
Additional Notes: This work is part of the author’s MSc thesis written under the supervision of Wiesław Kubiś. This research was supported by GAČR project 16-34860L and partially supported by MOBILITY project 7AMB15AT035 (RVO:67985840).
Communicated by: Mirna Džamonja
Article copyright: © Copyright 2017 American Mathematical Society