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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A rigid Urysohn-like metric space
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by Jan Grebík PDF
Proc. Amer. Math. Soc. 145 (2017), 4049-4060 Request permission

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.

References
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Additional Information
  • Jan Grebík
  • Affiliation: Institute of Mathematics, Czech Academy of Sciences, 115 67 Prague, Czech Republic
  • 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
  • © Copyright 2017 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 145 (2017), 4049-4060
  • MSC (2010): Primary 03C50, 05C63
  • DOI: https://doi.org/10.1090/proc/13511
  • MathSciNet review: 3665056