Nonstandard methods in Ramsey theory and combinatorial number theory

Preface......Page 7
Notation and Conventions......Page 11
Contents......Page 12
Part I Preliminaries......Page 16
  1.1 Basics on Ultrafilters......Page 17
  1.2 The Space of Ultrafilters βS......Page 19
  1.3 The Case of a Semigroup......Page 21
  1.5 Partial Semigroups......Page 22
  Notes and References......Page 23
  2.1 Warming-Up......Page 25
  2.2 The Star Map and the Transfer Principle......Page 28
   2.2.1 Additional Assumptions......Page 32
  2.3 The Transfer Principle, in Practice......Page 34
  2.4 The Ultrapower Model......Page 36
   2.4.1 The Ultrapower Construction......Page 37
   2.4.2 Hyper-Extensions in the Ultrapower Model......Page 38
   2.4.3 The Properness Condition in the Ultrapower Model......Page 41
   2.4.4 An Algebraic Presentation......Page 42
  2.5 Internal and External Objects......Page 43
   2.5.1 Internal Objects in the Ultrapower Model......Page 46
  2.6 Hyperfinite Sets......Page 47
   2.6.1 Hyperfinite Sums......Page 50
   2.7.1 An Application to Graph Theory......Page 51
  2.8 The Saturation Principle......Page 53
   2.8.1 Saturation in the Ultrapower Model......Page 54
  2.9 Hyperfinite Approximation......Page 55
  Notes and References......Page 56
  3.1 Hyperfinite Generators......Page 58
  3.2 The Case of a Semigroup Again......Page 60
  Notes and References......Page 61
  4.1 The Foundational Perspective......Page 62
  4.2 Revisiting Hyperfinite Generators......Page 64
  4.3 The Iterated Ultrapower Perspective......Page 65
  4.4 Revisiting Idempotents......Page 66
  Notes and References......Page 67
  5.1 Premeasures and Measures......Page 68
  5.2 The Definition of Loeb Measure......Page 70
  5.3 Lebesgue Measure via Loeb Measure......Page 72
  5.4 Integration......Page 73
  5.5 Product Measure......Page 78
  5.6 Ergodic Theory of Hypercycle Systems......Page 79
  Notes and References......Page 83
Part II Ramsey Theory......Page 84
  6.1 Infinite Ramsey\'s Theorem......Page 85
  6.3 Rado\'s Path Decomposition Theorem......Page 87
  6.4 Ultrafilter Trees......Page 88
  Notes and References......Page 91
  7.1 The Theorem of van der Waerden......Page 92
  7.2 The Hales-Jewett Theorem......Page 95
  Notes and References......Page 99
 8 From Hindman to Gowers......Page 100
  8.1 Hindman\'s Theorem......Page 101
  8.2 The Milliken-Taylor Theorem......Page 103
  8.3 Gowers\' Theorem......Page 104
  Notes and References......Page 109
 9 Partition Regularity of Equations......Page 110
  9.1 Characterizations of Partition Regularity......Page 111
  9.2 Rado\'s Theorem......Page 112
  9.3 Nonlinear Diophantine Equations: Some Examples......Page 115
  Notes and References......Page 117
Part III Combinatorial Number Theory......Page 120
  10.1 Densities......Page 121
  10.2 Structural Properties......Page 124
  10.3 Working in Z......Page 127
  10.4 Furstenberg\'s Correspondence Principle......Page 130
  Notes and References......Page 132
  11.1 Finite Embeddability Between Sets of Natural Numbers......Page 133
  11.2 Banach Density as Shnirelmann Density in the Remote Realm......Page 136
  11.3 Applications......Page 138
  Notes and References......Page 141
  12.1 The Statement of Jin\'s Sumset Theorem and Some Standard Consequences......Page 142
  12.2 Jin\'s Proof of the Sumset Theorem......Page 144
  12.3 Beiglböck\'s Proof......Page 146
  12.4 A Proof with an Explicit Bound......Page 148
  12.5 Quantitative Strengthenings......Page 149
  Notes and References......Page 152
  13.1 Erdős\' Conjecture......Page 153
  13.2 A 1-Shift Version of Erdős\' Conjecture......Page 156
  13.3 A Weak Density Version of Folkman\'s Theorem......Page 159
  Notes and References......Page 160
  14.1 The Main Theorem......Page 161
  14.2 Connection to the Erdős-Turán Conjecture......Page 165
  Notes and References......Page 168
  15.1 IM Sets......Page 169
  15.2 SIM Sets......Page 173
  Notes and References......Page 177
Part IV Other Topics......Page 178
  16.1 Triangle Removal Lemma......Page 179
  16.2 Szemerédi Regularity Lemma......Page 183
  Notes and References......Page 186
  17.1 Statement of Definitions and the Main Theorem......Page 187
  17.2 A Special Case: Approximate Groups of Finite Exponent......Page 189
  Notes and References......Page 192
  A.1.1 Mathematical Universes and Superstructures......Page 193
  A.1.2 Bounded Quantifier Formulas......Page 195
  A.1.3 Łos\' Theorem......Page 198
  A.1.4 Models That Allow Iterated Hyper-Extensions......Page 200
References......Page 202
Index......Page 208