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
