Ramsey Methods in Analysis

Foreword vii

A Saturated and Conditional Structures in Banach Spaces

Spiros A. Argyros 1

Introduction 3

I. Tsirelson and Mixed Tsirelson Spaces 7

II. Tree Complete Extensions of a Ground Norm 21

II.1 Mixed Tsirelson Extension of a Ground Norm . . . . . . . . . . . . 21

II.2 R.I.S. Sequences and the Basic Inequality . . . . . . . . . . . . . . 26

III. Hereditarily Indecomposable Extensions with a Schauder Basis 39

III.1 The HI Property in X[G, σ] . . . . . . . . . . . . . . . . . . . . . . 39

III.2 The HI Property in X[G, σ]∗ . . . . . . . . . . . . . . . . . . . . . . 43

IV. The Space of the Operators for HI Banach Spaces 47

IV.1 Some General Properties of HI Spaces . . . . . . . . . . . . . . . . 47

IV.2 The Space of Operators L(X[G, σ]), L(X[G, σ]∗) . . . . . . . . . . . 52

V. Examples of Hereditarily Indecomposable Extensions 57

V.1 A Quasi-reflexive HI Space . . . . . . . . . . . . . . . . . . . . . . 57

V.2 The Spaces p, 1 < p < ∞, are Quotients of HI Spaces . . . . . . . 58

V.3 A Non Separable HI Space . . . . . . . . . . . . . . . . . . . . . . . 62

VI. The Space Xω1 71

VII. Finite Representability of JT0 and the Diagonal Space D(Xγ) 81

VIII. The Spaces of Operators L(Xγ), L(X,Xω1) 87

Appendix A. Transfinite Schauder Basic Sequences 99

Appendix B. The Proof of the Finite Representability of JT0 105

Bibliography 117

vi Contents

B High-Dimensional Ramsey Theory and

Banach Space Geometry

Stevo Todorcevic 121

Introduction 123

I. Finite-Dimensional Ramsey Theory 127

I.1 Finite-Dimensional Ramsey Theorem . . . . . . . . . . . . . . . . . 127

I.2 SpreadingModels of Banach Spaces . . . . . . . . . . . . . . . . . 130

I.3 Finite Representability of Banach Spaces . . . . . . . . . . . . . . . 135

II. Ramsey Theory of Finite and Infinite Sequences 143

II.1 The Theory ofWell-Quasi-Ordered Sets . . . . . . . . . . . . . . . 143

II.2 Nash–Williams’ Theory of Fronts and Barriers . . . . . . . . . . . 147

II.3 UniformFronts and Barriers . . . . . . . . . . . . . . . . . . . . . . 153

II.4 Canonical Equivalence Relations on Uniform Fronts and Barriers . 165

II.5 Unconditional Subsequences of Weakly Null Sequences . . . . . . . 169

II.6 Topological Ramsey Theory . . . . . . . . . . . . . . . . . . . . . . 177

II.7 The Theory of Better-Quasi-Orderings . . . . . . . . . . . . . . . . 180

II.8 Ellentuck’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 185

II.9 Summability in Banach Spaces . . . . . . . . . . . . . . . . . . . . 188

II.10 Summability in Topological Abelian Groups . . . . . . . . . . . . . 192

III. Ramsey Theory of Finite and Infinite Block Sequences 197

III.1 Hindman’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 197

III.2 Canonical Equivalence Relations on FIN . . . . . . . . . . . . . . . 200

III.3 Fronts and Barriers on FIN[<∞] . . . . . . . . . . . . . . . . . . . . 201

III.4 Milliken’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

III.5 An Approximate Ramsey Theorem . . . . . . . . . . . . . . . . . . 209

IV. Approximate and Strategic Ramsey Theory of Banach Spaces 217

IV.1 Gowers’ Dichotomy . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

IV.2 Approximate and Strategic Ramsey Sets . . . . . . . . . . . . . . . 220

IV.3 Combinatorial Forcing on Block Sequences in Banach Spaces . . . 224

IV.4 Coding into Approximate and Strategic Ramsey Sets . . . . . . . . 229

IV.5 Topological Ramsey Theory of Block Sequences in Banach Spaces . 233

IV.6 An Application to Rough Classification of Banach Spaces . . . . . 240

IV.7 An Analytic Set whose Complement is not Approximately Ramsey 243

Bibliography 247

Index 253