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Structural Ramsey theory of metric spaces and topological dynamics of isometry groups


About this Title

L. Nguyen Van Thé, Department of Mathematics and Statistics, University of Calgary, 2500 University Drive NW, Calgary, Alberta, Canada, T2N1N4

Publication: Memoirs of the American Mathematical Society
Publication Year 2010: Volume 206, Number 968
ISBNs: 978-0-8218-4711-4 (print); 978-1-4704-0582-3 (online)
DOI: http://dx.doi.org/10.1090/S0065-9266-10-00586-7
Published electronically: February 19, 2010
MathSciNet review: 2667917
Keywords:Ramsey theory, Metric geometry, Fraïssé theory, Topological groups actions, Extreme amenability, Universal minimal flows, Oscillation stability, Urysohn metric space.
MSC (2000): Primary 03E02; Secondary 05C55, 05D10, 37B05, 46B99, 51F99

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Table of Contents


Chapters

  • Preamble
  • Preliminary remarks
  • Introduction
  • Chapter 1. Fraïssé classes of finite metric spaces and Urysohn spaces
  • Chapter 2. Ramsey calculus, Ramsey degrees and universal minimal flows
  • Chapter 3. Big Ramsey degrees, indivisibility and oscillation stability
  • Appendix A. Amalgamation classes $\mathcal {M}_S$ when $|S| \le 4$
  • Appendix B. Indivisibility of $\mathbf {U}_S$ when $|S| \le 4$
  • Appendix C. On the universal Urysohn space $\mathbf {U}$

Abstract


In 2003, Kechris, Pestov and Todorcevic showed that the structure of certain separable metric spaces - called ultrahomogeneous - is closely related to the combinatorial behavior of the class of their finite metric spaces. The purpose of the present paper is to explore different aspects of this connection.

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