A rigid Urysohn-like metric space
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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.
References
- Itaï Ben Yaacov, The linear isometry group of the Gurarij space is universal, Proc. Amer. Math. Soc. 142 (2014), no. 7, 2459–2467. MR 3195767, DOI 10.1090/S0002-9939-2014-11956-3
- Wojciech Bielas, An example of a rigid $\kappa$-superuniversal metric space, Topology Appl. 208 (2016), 127–142. MR 3506974, DOI 10.1016/j.topol.2016.05.010
- Stephen H. Hechler, Large superuniversal metric spaces, Israel J. Math. 14 (1973), 115–148. MR 324617, DOI 10.1007/BF02762669
- Wilfrid Hodges, Model theory, Encyclopedia of Mathematics and its Applications, vol. 42, Cambridge University Press, Cambridge, 1993. MR 1221741, DOI 10.1017/CBO9780511551574
- Wilfried Imrich, Sandi Klavžar, and Vladimir Trofimov, Distinguishing infinite graphs, Electron. J. Combin. 14 (2007), no. 1, Research Paper 36, 12. MR 2302543
- Wiesław Kubiś, Fraïssé sequences: category-theoretic approach to universal homogeneous structures, Ann. Pure Appl. Logic 165 (2014), no. 11, 1755–1811. MR 3244668, DOI 10.1016/j.apal.2014.07.004
- W. Kubiś and D. Mašulović, Katětov functors, Appl. Categor. Struct. (2016). doi:10.1007/s10485-016-9461-z
- R. Rado, Universal graphs and universal functions, Acta Arith. 9 (1964), 331–340. MR 172268, DOI 10.4064/aa-9-4-331-340
- Saharon Shelah, On universal graphs without instances of CH, Ann. Pure Appl. Logic 26 (1984), no. 1, 75–87. MR 739914, DOI 10.1016/0168-0072(84)90042-3
- P.S. Urysohn, Sur un espace métrique universel, I, II, Bull. Sci. Math. (2) 51 (1927) 43–64, 74–90
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