Ramsey theory is a fast-growing area of combinatorics with deep connections to other fields of mathematics such as topological dynamics, ergodic theory, mathematical logic, and algebra. The area of Ramsey theory dealing with Ramsey-type phenomena in higher dimensions is particularly useful. Introduction to Ramsey Spaces presents in a systematic way a method for building higher-dimensional Ramsey spaces from basic one-dimensional principles. It is the first book-length treatment of this area of Ramsey theory, and emphasizes applications for related and surrounding fields of mathematics, such as set theory, combinatorics, real and functional analysis, and topology. In order to facilitate accessibility, the book gives the method in its axiomatic form with examples that cover many important parts of Ramsey theory both finite and infinite. An exciting new direction for combinatorics, this book will interest graduate students and researchers working in mathematical subdisciplines requiring the mastery and practice of high-dimensional Ramsey theory.

Introduction 1Chapter 1. Ramsey Theory: Preliminaries 3
1.1 Coideals 3
1.2 Dimensions in Ramsey Theory 5
1.3 Higher Dimensions in Ramsey Theory 10
1.4 Ramsey Property and Baire Property 20


Chapter 2. Semigroup Colorings 27
2.1 Idempotents in Compact semigroups 27
2.2 The Galvin-Glazer Theorem 30
2.3 Gowers's Theorem 34
2.4 A Semigroup of Subsymmetric Ultrafilters 38
2.5 The Hales-Jewett Theorem 41
2.6 Partial Semigroup of Located Words 46


Chapter 3. Trees and Products 49
3.1 Versions of the Halpern-Lauchli Theorem 49
3.2 A Proof of the Halpern-Lauchli Theorem 55
3.3 Products of Finite Sets 57


Chapter 4. Abstract Ramsey Theory 63
4.1 Abstract Baire Property 63
4.2 The Abstract Ramsey Theorem 68
4.3 Combinatorial Forcing 76
4.4 The Hales-Jewett Space 83
4.5 Ramsey Spaces of Infinite Block Sequences of Located Words 89


Chapter 5. Topological Ramsey Theory 93
5.1 Topological Ramsey Spaces 93
5.2 Topological Ramsey Spaces of Infinite Block Sequences of Vectors 99
5.3 Topological Ramsey Spaces of Infinite Sequences of Variable Words 105
5.4 Parametrized Versions of Rosenthal Dichotomies 111
5.5 Ramsey Theory of Superperfect Subsets of Polish Spaces 117
5.6 Dual Ramsey Theory 121
5.7 A Ramsey Space of Infinite-Dimensional Vector Subspaces of FN 127


Chapter 6. Spaces of Trees 135
6.1 A Ramsey Space of Strong Subtrees 135
6.2 Applications of the Ramsey Space of Strong Subtrees 138
6.3 Partition Calculus on Finite Powers of the Countable Dense Linear Ordering 143
6.4 A Ramsey Space of Increasing Sequences of Rationals 149
6.5 Continuous Colorings on Q[k] 152
6.6 Some Perfect Set Theorems 158
6.7 Analytic Ideals and Points in Compact Sets of the First Baire Class 165


Chapter 7. Local Ramsey Theory 179
7.1 Local Ellentuck Theory 179
7.2 Topological Ultra-Ramsey Spaces 190
7.3 Some Examples of Selective Coideals on N 194
7.4 Some Applications of Ultra-Ramsey Theory 198
7.5 Local Ramsey Theory and Analytic Topologies on N 202
7.6 Ultra-Hales-Jewett Spaces 207
7.7 Ultra-Ramsey Spaces of Block Sequences of Located Words 212
7.8 Ultra-Ramsey Space of Infinite Block Sequences of Vectors 215


Chapter 8. Infinite Products of Finite Sets 219
8.1 Semicontinuous Colorings of Infinite Products of Finite Sets 219
8.2 Polarized Ramsey Property 224
8.3 Polarized Partition Calculus 231


Chapter 9. Parametrized Ramsey Theory 237
9.1 Higher Dimensional Ramsey Theorems
Parametrized by Infinite Products of Finite Sets 237
9.2 Combinatorial Forcing Parametrized by Infinite Products of Finite Sets 243
9.3 Parametrized Ramsey Property 248
9.4 Infinite-Dimensional Ramsey Theorem Parametrized by Infinite Products of Finite Sets 254


Appendix 259
Bibliography 271
Subject Index 279
Index of Notation 285