Introduction to Ramsey Spaces

✍ Scribed by Stevo Todorcevic

• Publisher: Princeton University Press
• Year: 2010
• Tongue: English
• Leaves: 296
• Series: Annals of Mathematics Studies 174
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Title......Page 4
Copyright......Page 5
Contents......Page 6
Introduction......Page 10
1.1 Coideals......Page 12
1.2 Dimensions in Ramsey Theory......Page 14
1.3 Higher Dimensions in Ramsey Theory......Page 19
1.4 Ramsey Property and Baire Property......Page 29
2.1 Idempotents in Compact semigroups......Page 36
2.2 The Galvin-Glazer Theorem......Page 39
2.3 Gowers's Theorem......Page 43
2.4 A Semigroup of Subsymmetric Ultrafilters......Page 47
2.5 The Hales-Jewett Theorem......Page 50
2.6 Partial Semigroup of Located Words......Page 55
3.1 Versions of the Halpern-Läuchli Theorem......Page 58
3.2 A Proof of the Halpern-Läuchli Theorem......Page 64
3.3 Products of Finite Sets......Page 66
4.1 Abstract Baire Property......Page 72
4.2 The Abstract Ramsey Theorem......Page 77
4.3 Combinatorial Forcing......Page 85
4.4 The Hales-Jewett Space......Page 92
4.5 Ramsey Spaces of Infinite Block Sequences of Located Words......Page 98
5.1 Topological Ramsey Spaces......Page 102
5.2 Topological Ramsey Spaces of Infinite Block Sequences of Vectors......Page 108
5.3 Topological Ramsey Spaces of Infinite Sequences of Variable Words......Page 114
5.4 Parametrized Versions of Rosenthal Dichotomies......Page 120
5.5 Ramsey Theory of Superperfect Subsets of Polish Spaces......Page 126
5.6 Dual Ramsey Theory......Page 130
5.7 A Ramsey Space of Infinite-Dimensional Vector Subspaces of F[sup(N)]......Page 136
6.1 A Ramsey Space of Strong Subtrees......Page 144
6.2 Applications of the Ramsey Space of Strong Subtrees......Page 147
6.3 Partition Calculus on Finite Powers of the Countable Dense Linear Ordering......Page 152
6.4 A Ramsey Space of Increasing Sequences of Rationals......Page 158
6.5 Continuous Colorings on Q[sup([k])]......Page 161
6.6 Some Perfect Set Theorems......Page 167
6.7 Analytic Ideals and Points in Compact Sets of the First Baire Class......Page 174
7.1 Local Ellentuck Theory......Page 188
7.2 Topological Ultra-Ramsey Spaces......Page 199
7.3 Some Examples of Selective Coideals on N......Page 203
7.4 Some Applications of Ultra-Ramsey Theory......Page 207
7.5 Local Ramsey Theory and Analytic Topologies on N......Page 211
7.6 Ultra-Hales-Jewett Spaces......Page 216
7.7 Ultra-Ramsey Spaces of Block Sequences of Located Words......Page 221
7.8 Ultra-Ramsey Space of Infinite Block Sequences of Vectors......Page 224
8.1 Semicontinuous Colorings of Infinite Products of Finite Sets......Page 228
8.2 Polarized Ramsey Property......Page 233
8.3 Polarized Partition Calculus......Page 240
9.1 Higher Dimensional Ramsey Theorems Parametrized by Infinite Products of Finite Sets......Page 246
9.2 Combinatorial Forcing Parametrized by Infinite Products of Finite Sets......Page 252
9.3 Parametrized Ramsey Property......Page 257
9.4 Infinite-Dimensional Ramsey Theorem Parametrized by Infinite Products of Finite Sets......Page 263
Appendix......Page 268
Bibliography......Page 280
C......Page 288
H......Page 289
O......Page 290
S......Page 291
U......Page 292
Z......Page 293
Index of Notation......Page 294