A survey on structural Ramsey theory and topological dynamics with the Kechris-Pestov-Todorcevic correspondence in mind

The article [KPT05], published in 2005 by Kechris, Pestov and Todorcevic, established a surprising correspondence between structural Ramsey theory and topological dynamics. As an immediate consequence, it triggered a new interest for structural Ramsey theory. The purpose of the present paper is to present a selfcontained survey of the corresponding developments. The paper is organized as follows: Section 2 covers the fundamentals of structural Ramsey theory and of Fräıssé theory. Section 3 introduces the necessary notions from dynamics. Section 4 presents the Kechris-Pestov-Todorcevic (in short, KPT) correspondence. Section 5 indicates a list of situations where the correspondence was successfully applied. Finally, Section 6 contains a list of open problems. Most of the material presented here is based on [KPT05] as well as on the habilitation memoir [NVT13b].

[1]  B. Weiss,et al.  THE UNIVERSAL MINIMAL SYSTEM FOR THE GROUP OF HOMEOMORPHISMS OF THE CANTOR SET , 2003 .

[2]  J. Spencer.  Ramsey’s theorem for spaces , 1979 .

[3]  V. Pestov.  ON FREE ACTIONS, MINIMAL FLOWS, AND A PROBLEM BY ELLIS , 1998 .

[4]  Stevo Todorcevic,et al.  Introduction to Ramsey Spaces , 2010 .

[5]  Miodrag Sokic.  Ramsey Properties of Finite Posets , 2012, Order.

[6]  A. H. Lachlan,et al.  Countable homogeneous tournaments , 1984 .

[7]  Simon Thomas,et al.  Reducts of the random graph , 1991, Journal of Symbolic Logic.

[8]  Leo Harrington,et al.  Models Without Indiscernibles , 1978, J. Symb. Log..

[9]  Frank Plumpton Ramsey,et al.  On a Problem of Formal Logic , 1930 .

[10]  R. Woodrow,et al.  Countable ultrahomogeneous undirected graphs , 1980 .

[11]  M. Gromov,et al.  A topological application of the isoperimetric inequality , 1983 .

[12]  A. Kechris,et al.  Turbulence, amalgamation, and generic automorphisms of homogeneous structures , 2004, math/0409567.

[13]  An Infinite self dual Ramsey theorem , 2013, 1310.3599.

[14]  A. Kechris.  Dynamics of non-archimedean Polish groups , 2012 .

[15]  Russell Lyons,et al.  Random Orderings and Unique Ergodicity of Automorphism Groups , 2012, 1208.2389.

[16]  R L Graham,et al.  Ramsey's Theorem for a Class of Categories. , 1972, Proceedings of the National Academy of Sciences of the United States of America.

[17]  Dynamical properties of the automorphism groups of the random poset and random distributive lattice , 2012 .

[18]  S. Solecki,et al.  Projective Fraïssé limits and the pseudo-arc , 2006 .

[19]  Claude Laflamme,et al.  Partition properties of the dense local order and a colored version of Milliken’s theorem , 2010, Comb..

[20]  Manuel Bodirsky.  New Ramsey Classes from Old , 2014, Electron. J. Comb..

[21]  Benjamin Weiss,et al.  Minimal actions of the group $ {\Bbb S(Z)} $ of permutations of the integers , 2002 .

[22]  Michael Pinsker,et al.  Reducts of Ramsey structures , 2011, AMS-ASL Joint Special Session.

[23]  R. Graham,et al.  Ramsey’s theorem for $n$-parameter sets , 1971 .

[24]  Alexander S. Kechris,et al.  The Descriptive Set Theory of Polish Group Actions: DESCRIPTIVE SET THEORY , 1996 .

[25]  L. Nguyen Van Th'e,et al.  Structural Ramsey theory of metric spaces and topological dynamics of isometry groups , 2008, 0804.1593.

[26]  Dana Bartovsov'a.  Universal Minimal Flows of Groups of Automorphisms of Uncountable Structures , 2011, 1111.4995.

[27]  Michael Pinsker,et al.  Decidability of Definability , 2013, The Journal of Symbolic Logic.

[28]  Robert E. Woodrow.  Theories with a Finite Number of Countable Models , 1978, J. Symb. Log..

[29]  Vojtech Rödl,et al.  Partitions of Finite Relational and Set Systems , 1977, J. Comb. Theory A.

[30]  W. T. Trotter,et al.  Graphs and Orders in Ramsey Theory and in Dimension Theory , 1985 .

[31]  G. Cherlin,et al.  The Classification of Countable Homogeneous Directed Graphs and Countable Homogeneous N-Tournaments , 1998 .

[32]  S. Solecki,et al.  Extending partial isometries , 2005 .

[33]  Julia Böttcher,et al.  Ramsey Properties of Permutations , 2013, Electron. J. Comb..

[34]  Claude Laflamme,et al.  Ramsey Precompact Expansions of Homogeneous Directed Graphs , 2013, Electron. J. Comb..

[35]  Ramsey Theory,et al.  Ramsey Theory , 2020, Set Theory and Foundations of Mathematics: An Introduction to Mathematical Logic.

[36]  Daniel McDonald,et al.  Automorphism Groups of Countably Categorical Linear Orders are Extremely Amenable , 2013, Order.

[37]  Julien Melleray,et al.  Extremely amenable groups via continuous logic , 2014, 1404.4590.