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Introduction to Ramsey Spaces (AM-174) : Annals of Mathematics Studies (Paperback) - Stevo Todorcevic

Introduction to Ramsey Spaces (AM-174)

Annals of Mathematics Studies (Paperback)

Paperback Published: 21st July 2010
ISBN: 9780691145426
Number Of Pages: 287

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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.

Introductionp. 1
Ramsey Theory: Preliminariesp. 3
Coidealsp. 3
Dimensions in Ramsey Theoryp. 5
Higher Dimensions in Ramsey Theoryp. 10
Ramsey Property and Baire Propertyp. 20
Semigroup Coloringsp. 27
Idempotents in Compact semigroupsp. 27
The Galvin-Glazer Theoremp. 30
Gowers's Theoremp. 34
A Semigroup of Subsymmetric Ultrafiltersp. 38
The Hales-Jewett Theoremp. 41
Partial Semigroup of Located Wordsp. 46
Trees and Productsp. 49
Versions of the Halpern-Läuchli Theoremp. 49
A Proof of the Halpern-Läuchli Theoremp. 55
Products of Finite Setsp. 57
Abstract Ramsey Theoryp. 63
Abstract Baire Propertyp. 63
The Abstract Ramsey Theoremp. 68
Combinatorial Forcingp. 76
The Hales-Jewett Spacep. 83
Ramsey Spaces of Infinite Block Sequences of Located Wordsp. 89
Topological Ramsey Theoryp. 93
Topological Ramsey Spacesp. 93
Topological Ramsey Spaces of Infinite Block Sequences of Vectorsp. 99
Topological Ramsey Spaces of Infinite Sequences of Variable Wordsp. 105
Parametrized Versions of Rosenthal Dichotomiesp. 111
Ramsey Theory of Superperfect Subsets of Polish Spacesp. 117
Dual Ramsey Theoryp. 121
A Ramsey Space of Infinite-Dimensional Vector Subspaces of FNp. 127
Spaces of Treesp. 135
A Ramsey Space of Strong Subtreesp. 135
Applications of the Ramsey Space of Strong Subtreesp. 138
Partition Calculus on Finite Powers of the Countable Dense Linear Orderingp. 143
A Ramsey Space of Increasing Sequences of Rationalsp. 149
Continuous Colorings on Q[k]p. 152
Some Perfect Set Theoremsp. 158
Analytic Ideals and Points in Compact Sets of the First Baire Classp. 165
Local Ramsey Theoryp. 179
Local Ellentuck Theoryp. 179
Topological Ultra-Ramsey Spacesp. 190
Some Examples of Selective Coideals on Np. 194
Some Applications of Ultra-Ramsey Theoryp. 198
Local Ramsey Theory and Analytic Topologies on Np. 202
Ultra-Hales-Jewett Spacesp. 207
Ultra-Ramsey Spaces of Block Sequences of Located Wordsp. 212
Ultra-Ramsey Space of Infinite Block Sequences of Vectorsp. 215
Infinite Products of Finite Setsp. 219
Semicontinuous Colorings of Infinite Products of Finite Setsp. 219
Polarized Ramsey Propertyp. 224
Polarized Partition Calculusp. 231
Parametrized Ramsey Theoryp. 237
Higher Dimensional Ramsey Theorems Parametrized by Infinite Products of Finite Setsp. 237
Combinatorial Forcing Parametrized by Infinite Products of Finite Setsp. 243
Parametrized Ramsey Propertyp. 248
Infinite-Dimensional Ramsey Theorem Parametrized by Infinite Products of Finite Setsp. 254
Appendixp. 259
Bibliographyp. 271
Subject Indexp. 279
Index of Notationp. 285
Table of Contents provided by Ingram. All Rights Reserved.

ISBN: 9780691145426
ISBN-10: 0691145423
Series: Annals of Mathematics Studies (Paperback)
Audience: Professional
Format: Paperback
Language: English
Number Of Pages: 287
Published: 21st July 2010
Country of Publication: US
Dimensions (cm): 23.11 x 15.49  x 1.78
Weight (kg): 0.43

Earn 165 Qantas Points
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