Memoirs of the American Mathematical Society 2010; 140 pp; softcover Volume: 206 ISBN10: 0821847112 ISBN13: 9780821847114 List Price: US$76 Individual Members: US$45.60 Institutional Members: US$60.80 Order Code: MEMO/206/968
 In 2003, Kechris, Pestov and Todorcevic showed that the structure of certain separable metric spacescalled ultrahomogeneousis 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. Table of Contents  Introduction
 Fraïssé classes of finite metric spaces and Urysohn spaces
 Ramsey calculus, Ramsey degrees and universal minimal flows
 Big Ramsey degrees, indivisibility and oscillation stability
 Appendix A. Amalgamation classes \(\mathcal {M}_S\) when \(S \leqslant 4\)
 Appendix B. Indivisibility of \(\mathbf {U}_S\) when \(S \leqslant 4\)
 Appendix C. On the universal Urysohn space \(\mathbf {U}\)
 Bibliography
 Index
