内容

目次

Introduction 1 Chapter 1. Ramsey Theory: Preliminaries 3 1.1 Coideals3 1.2 Dimensions in Ramsey Theory 5 1.3 Higher Dimensions in Ramsey Theory10 1.4 Ramsey Property and Baire Property 20 Chapter 2. SemigroupColorings 27 2.1 Idempotents in Compact semigroups 27 2.2 The Galvin-GlazerTheorem 30 2.3 Gowers's Theorem 34 2.4 A Semigroup of SubsymmetricUltrafilters 38 2.5 The Hales-Jewett Theorem 41 2.6 Partial Semigroup ofLocated Words 46 Chapter 3. Trees and Products 49 3.1 Versions of theHalpern-Lauchli Theorem 49 3.2 A Proof of the Halpern-Lauchli Theorem 553.3 Products of Finite Sets 57 Chapter 4. Abstract Ramsey Theory 63 4.1Abstract Baire Property 63 4.2 The Abstract Ramsey Theorem 68 4.3Combinatorial Forcing 76 4.4 The Hales-Jewett Space 83 4.5 Ramsey Spaces ofInfinite Block Sequences of Located Words 89 Chapter 5. TopologicalRamsey Theory 93 5.1 Topological Ramsey Spaces 93 5.2 Topological RamseySpaces of Infinite Block Sequences of Vectors 99 5.3 Topological RamseySpaces of Infinite Sequences of Variable Words 105 5.4 Parametrized Versionsof Rosenthal Dichotomies 111 5.5 Ramsey Theory of Superperfect Subsets ofPolish Spaces 117 5.6 Dual Ramsey Theory 121 5.7 A Ramsey Space ofInfinite-Dimensional Vector Subspaces of FN 127 Chapter 6. Spaces ofTrees 135 6.1 A Ramsey Space of Strong Subtrees 135 6.2 Applications of theRamsey Space of Strong Subtrees 138 6.3 Partition Calculus on Finite Powersof the Countable Dense Linear Ordering 143 6.4 A Ramsey Space of IncreasingSequences of Rationals 149 6.5 Continuous Colorings on Q[k] 152 6.6 SomePerfect Set Theorems 158 6.7 Analytic Ideals and Points in Compact Sets ofthe First Baire Class 165 Chapter 7. Local Ramsey Theory 179 7.1 LocalEllentuck Theory 179 7.2 Topological Ultra-Ramsey Spaces 190 7.3 SomeExamples of Selective Coideals on N 194 7.4 Some Applications ofUltra-Ramsey Theory 198 7.5 Local Ramsey Theory and Analytic Topologies on N202 7.6 Ultra-Hales-Jewett Spaces 207 7.7 Ultra-Ramsey Spaces of BlockSequences of Located Words 212 7.8 Ultra-Ramsey Space of Infinite BlockSequences of Vectors 215 Chapter 8. Infinite Products of Finite Sets 2198.1 Semicontinuous Colorings of Infinite Products of Finite Sets 219 8.2Polarized Ramsey Property 224 8.3 Polarized Partition Calculus 231Chapter 9. Parametrized Ramsey Theory 237 9.1 Higher Dimensional RamseyTheorems Parametrized by Infinite Products of Finite Sets 237 9.2Combinatorial Forcing Parametrized by Infinite Products of Finite Sets 2439.3 Parametrized Ramsey Property 248 9.4 Infinite-Dimensional Ramsey TheoremParametrized by Infinite Products of Finite Sets 254 Appendix 259Bibliography 271 Subject Index 279 Index of Notation 285