Canonical Ramsey Theory on Polish Spaces (Cambridge Tracts in Mathematics)

[Book Description]

This book lays the foundations for an exciting new area of research in descriptive set theory. It develops a robust connection between two active topics: forcing and analytic equivalence relations. This in turn allows the authors to develop a generalization of classical Ramsey theory. Given an analytic equivalence relation on a Polish space, can one find a large subset of the space on which it has a simple form? The book provides many positive and negative general answers to this question. The proofs feature proper forcing and Gandy-Harrington forcing, as well as partition arguments. The results include strong canonization theorems for many classes of equivalence relations and sigma-ideals, as well as ergodicity results in cases where canonization theorems are impossible to achieve. Ideal for graduate students and researchers in set theory, the book provides a useful springboard for further research.

[Table of Contents]

Preface                                            vii
    1 Introduction                                 1  (13)
      1.1 Motivation                               1  (2)
      1.2 Basic concepts                           3  (4)
      1.3 Outline of results                       7  (3)
      1.4 Navigation                               10 (2)
      1.5 Notation                                 12 (2)
    2 Background facts                             14 (31)
      2.1 Descriptive set theory                   14 (2)
      2.2 Invariant descriptive set theory         16 (2)
      2.3 Forcing                                  18 (5)
      2.4 Absoluteness and interpretations         23 (8)
      2.5 Generic ultrapowers                      31 (2)
      2.6 Idealized forcing                        33 (9)
      2.7 Concentration of measure                 42 (1)
      2.8 Katetov order and coding functions       42 (3)
    3 Analytic equivalence relations and models    45 (17)
    of set theory
      3.1 The model V[x]E                          45 (9)
      3.2 The model V [[x]]E                       54 (2)
      3.3 The poset P1E                            56 (6)
    4 Classes of equivalence relations             62 (18)
      4.1 Smooth equivalence relations             62 (1)
      4.2 Countable equivalence relations          63 (5)
      4.3 Equivalence relations classifiable by    68 (8)
      countable structures
      4.4 Hypersmooth equivalence relations        76 (2)
      4.5 Analytic vs. Borel equivalence           78 (2)
      relations
    5 Games and the Silver property                80 (20)
      5.1 Integer games connected with             80 (6)
      σ-ideals
      5.2 Determinacy conclusions                  86 (4)
      5.3 The selection property                   90 (7)
      5.4 A Silver-type dichotomy for a            97 (3)
      σ-ideal
    6 The game ideals                              100(47)
      6.1 σ-ideals σ-generated by      100(20)
      closed sets
      6.2 Porosity ideals                          120(14)
      6.3 Laver forcing                            134(2)
      6.4 Fat tree forcings                        136(3)
      6.5 The Lebesgue null ideal                  139(8)
    7 Benchmark equivalence relations              147(40)
      7.1 The E0 ideal                             147(3)
      7.2 The E1 ideal                             150(22)
      7.3 The E2 ideal                             172(15)
    8 Ramsey-type ideals                           187(31)
      8.1 Halpern-Lauchli cubes                    187(3)
      8.2 Silver cubes                             190(6)
      8.3 Ellentuck cubes                          196(6)
      8.4 Milliken cubes                           202(5)
      8.5 Matet cubes                              207(11)
    9 Product-type ideals                          218(26)
      9.1 Finite products and the covering         218(4)
      property
      9.2 Finite products of the E0 ideal          222(10)
      9.3 Infinite products of perfect sets        232(8)
      9.4 Products of superperfect sets            240(4)
    10 The countable support iteration ideals      244(20)
      10.1 Iteration preliminaries                 245(2)
      10.2 The weak Sacks property                 247(1)
      10.3 A game reformulation                    248(7)
      10.4 The canonization result                 255(5)
      10.5 The iteration of Sacks forcing          260(1)
      10.6 Anticanonization results                261(3)
References                                         264(4)
Index                                              268