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Nonstandard Methods in Ramsey Theory and Combinatorial Number Theory 2019 ed. [Pehme köide]

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  • Formaat: Paperback / softback, 206 pages, kõrgus x laius: 235x155 mm, kaal: 454 g, 2 Illustrations, black and white; XVI, 206 p. 2 illus., 1 Paperback / softback
  • Sari: Lecture Notes in Mathematics 2239
  • Ilmumisaeg: 24-May-2019
  • Kirjastus: Springer Nature Switzerland AG
  • ISBN-10: 3030179559
  • ISBN-13: 9783030179557
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Nonstandard Methods in Ramsey Theory and Combinatorial Number Theory 2019 ed.Suurem pilt
  • Formaat: Paperback / softback, 206 pages, kõrgus x laius: 235x155 mm, kaal: 454 g, 2 Illustrations, black and white; XVI, 206 p. 2 illus., 1 Paperback / softback
  • Sari: Lecture Notes in Mathematics 2239
  • Ilmumisaeg: 24-May-2019
  • Kirjastus: Springer Nature Switzerland AG
  • ISBN-10: 3030179559
  • ISBN-13: 9783030179557
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9783030179557.html

The goal of this monograph is to give an accessible introduction to nonstandard methods and their applications, with an emphasis on combinatorics and Ramsey theory. It includes both new nonstandard proofs of classical results and recent developments initially obtained in the nonstandard setting. This makes it the first combinatorics-focused account of nonstandard methods to be aimed at a general (graduate-level) mathematical audience. This book will provide a natural starting point for researchers interested in approaching the rapidly growing literature on combinatorial results obtained via nonstandard methods. The primary audience consists of graduate students and specialists in logic and combinatorics who wish to pursue research at the interface between these areas.


Part I Preliminaries
1 Ultrafilters
3(8)
1.1 Basics on Ultrafilters
3(2)
1.2 The Space of Ultrafilters βS
5(2)
1.3 The Case of a Semigroup
7(1)
1.4 The Existence of Idempotents in Semitopological Semigroups
8(1)
1.5 Partial Semigroups
8(1)
Notes and References
9(2)
2 Nonstandard Analysis
11(34)
2.1 Warming-Up
11(3)
2.2 The Star Map and the Transfer Principle
14(6)
2.2.1 Additional Assumptions
18(2)
2.3 The Transfer Principle, in Practice
20(2)
2.4 The Ultrapower Model
22(7)
2.4.1 The Ultrapower Construction
23(1)
2.4.2 Hyper-Extensions in the Ultrapower Model
24(3)
2.4.3 The Properness Condition in the Ultrapower Model
27(1)
2.4.4 An Algebraic Presentation
28(1)
2.5 Internal and External Objects
29(4)
2.5.1 Internal Objects in the Ultrapower Model
32(1)
2.6 Hyperfinite Sets
33(4)
2.6.1 Hyperfinite Sums
36(1)
2.7 Overflow and Underflow Principles
37(2)
2.7.1 An Application to Graph Theory
37(2)
2.8 The Saturation Principle
39(2)
2.8.1 Saturation in the Ultrapower Model
40(1)
2.9 Hyperfinite Approximation
41(1)
Notes and References
42(3)
3 Hyperfinite Generators of Ultrafilters
45(4)
3.1 Hyperfinite Generators
45(2)
3.2 The Case of a Semigroup Again
47(1)
Notes and References
48(1)
4 Many Stars: Iterated Nonstandard Extensions
49(6)
4.1 The Foundational Perspective
49(2)
4.2 Revisiting Hyperfinite Generators
51(1)
4.3 The Iterated Ultrapower Perspective
52(1)
4.4 Revisiting Idempotents
53(1)
Notes and References
54(1)
5 Loeb Measure
55(18)
5.1 Premcasures and Measures
55(2)
5.2 The Definition of Loeb Measure
57(2)
5.3 Lebesgue Measure via Loeb Measure
59(1)
5.4 Integration
60(5)
5.5 Product Measure
65(1)
5.6 Ergodic Theory of Hypercycle Systems
66(4)
Notes and References
70(3)
Part II Ramsey Theory
6 Ramsey's Theorem
73(8)
6.1 Infinite Ramsey's Theorem
73(2)
6.2 Finite Ramsey Theorem
75(1)
6.3 Rado's Path Decomposition Theorem
75(1)
6.4 Ultrafilter Trees
76(3)
Notes and References
79(2)
7 The Theorems of van der Waerden and Hales-Jewett
81(8)
7.1 The Theorem of van der Waerden
81(3)
7.2 The Hales-Jewett Theorem
84(4)
Notes and References
88(1)
8 From Hindman to Gowers
89(10)
8.1 Hindman's Theorem
90(2)
8.2 The Milliken-Taylor Theorem
92(1)
8.3 Gowers' Theorem
93(5)
Notes and References
98(1)
9 Partition Regularity of Equations
99(12)
9.1 Characterizations of Partition Regularity
100(1)
9.2 Rado's Theorem
101(3)
9.3 Nonlinear Diophantine Equations: Some Examples
104(2)
Notes and References
106(5)
Part III Combinatorial Number Theory
10 Densities and Structural Properties
111(12)
10.1 Densities
111(3)
10.2 Structural Properties
114(3)
10.3 Working in Z
117(3)
10.4 Furstenberg's Correspondence Principle
120(2)
Notes and References
122(1)
11 Working in the Remote Realm
123(10)
11.1 Finite Embeddability Between Sets of Natural Numbers
123(3)
11.2 Banach Density as Shnirelmann Density in the Remote Realm
126(2)
11.3 Applications
128(3)
Notes and References
131(2)
12 Jin's Sumset Theorem
133(12)
12.1 The Statement of Jin's Sumset Theorem and Some Standard Consequences
133(2)
12.2 Jin's Proof of the Sumset Theorem
135(2)
12.3 Beiglbock's Proof
137(2)
12.4 A Proof with an Explicit Bound
139(1)
12.5 Quantitative Strengthenings
140(3)
Notes and References
143(2)
13 Sumset Configurations in Sets of Positive Density
145(8)
13.1 Erdos' Conjecture
145(3)
13.2 A 1-Shift Version of Erdos' Conjecture
148(3)
13.3 A Weak Density Version of Folkman's Theorem
151(1)
Notes and References
152(1)
14 Near Arithmetic Progressions in Sparse Sets
153(8)
14.1 The Main Theorem
153(4)
14.2 Connection to the Erdos-Turan Conjecture
157(3)
Notes and References
160(1)
15 The Interval Measure Property
161(12)
15.1 IM Sets
161(4)
15.2 SIM Sets
165(4)
Notes and References
169(4)
Part IV Other Topics
16 Triangle Removal and Szemeredi Regularity
173(8)
16.1 Triangle Removal Lemma
173(4)
16.2 Szemeredi Regularity Lemma
177(3)
Notes and References
180(1)
17 Approximate Groups
181(6)
17.1 Statement of Definitions and the Main Theorem
181(2)
17.2 A Special Case: Approximate Groups of Finite Exponent
183(3)
Notes and References
186(1)
A Foundations of Nonstandard Analysis 187(10)
A.1 Foundations
187(10)
A.1.1 Mathematical Universes and Superstructures
187(2)
A.1.2 Bounded Quantifier Formulas
189(3)
A.1.3 Los' Theorem
192(2)
A.1.4 Models That Allow Iterated Hyper-Extensions
194(3)
References 197(6)
Index 203