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E-raamat: Fundamentals of Ramsey Theory

(Colgate University)
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Fundamentals of Ramsey TheorySuurem pilt
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Ramsey theory is a fascinating topic. The author shares his view of the topic in this contemporary overview of Ramsey theory. He presents from several points of view, adding intuition and detailed proofs, in an accessible manner unique among most books on the topic. This book covers all of the main results in Ramsey theory along with results that have not appeared in a book before.

The presentation is comprehensive and reader friendly. The book covers integer, graph, and Euclidean Ramsey theory with many proofs being combinatorial in nature. The author motivates topics and discussion, rather than just a list of theorems and proofs. In order to engage the reader, each chapter has a section of exercises.

This up-to-date book introduces the field of Ramsey theory from several different viewpoints so that the reader can decide which flavor of Ramsey theory best suits them.

Additionally, the book offers:

  • A chapter providing different approaches to Ramsey theory, e.g., using topological dynamics, ergodic systems, and algebra in the Stone-Cech compactification of the integers.

  • A chapter on the probabilistic method since it is quite central to Ramsey-type numbers.

  • A unique chapter presenting some applications of Ramsey theory.

  • Exercises in every chapter

The intended audience consists of students and mathematicians desiring to learn about Ramsey theory. An undergraduate degree in mathematics (or its equivalent for advanced undergraduates) and a combinatorics course is assumed.

TABLE OF CONENTS

Preface

List of Figures

List of Tables

Symbols

1. Introduction

2. Integer Ramsey Theory

3. Graph Ramsey Theory

4. Euclidean Ramsey Theory

5. Other Approaches to Ramsey Theory

6. The Probabilistic Method

7. Applications

Bibliography

Index

Biography

Aaron Robertson

 received his Ph.D. in mathematics from Temple University under the guidance of his advisor Doron Zeilberger. Upon finishing his Ph.D. he started at Colgate University in upstate New York where he is currently Professor of Mathematics. He also serves as Associate Managing editor of the journal Integers

. After a brief detour into the world of permutation patterns, he has focused most of his research on Ramsey theory.

Preface xi
Symbols xiii
1 Introduction
1(14)
1.1 What is Ramsey Theory?
1(1)
1.2 Notations and Conventions
2(1)
1.3 Prerequisites
3(7)
1.3.1 Combinatorics
3(1)
1.3.2 Analysis
4(3)
1.3.3 Probability
7(1)
1.3.4 Algebra
7(1)
1.3.5 Topology
8(1)
1.3.6 Statistics
9(1)
1.3.7 Practice
9(1)
1.4 Compactness Principle
10(1)
1.5 Set Theoretic Considerations
10(1)
1.6 Exercises
11(4)
2 Integer Ramsey Theory
15(72)
2.1 Van der Waerden's Theorem
15(9)
2.1.1 Hilbert's Cube Lemma
20(2)
2.1.2 Deuber's Theorem
22(2)
2.2 Equations
24(21)
2.2.1 Schur's Theorem
24(3)
2.2.2 Rado's Theorem
27(5)
2.2.2.1 Some Rado Numbers
32(1)
2.2.3 Nonlinear Equations
33(11)
2.2.4 Algebraic Equations
44(1)
2.3 Hales-Jewett Theorem
45(6)
2.4 Finite Sums
51(10)
2.4.1 Arnautov-Folkman-Rado-Sanders' Theorem
52(3)
2.4.2 Hindman's Theorem
55(6)
2.5 Density Results
61(22)
2.5.1 Roth's Theorem
65(16)
2.5.2 Szemeredi's Theorem
81(1)
2.5.3 Density Hales-Jewett Theorem
82(1)
2.6 Exercises
83(4)
3 Graph Ramsey Theory
87(36)
3.1 Complete Graphs
87(7)
3.1.1 Deducing Schur's Theorem
93(1)
3.2 Other Graphs
94(5)
3.2.1 Some Graph Theory Concepts
95(1)
3.2.2 Graph Ramsey Numbers
96(3)
3.3 Hypergraphs
99(13)
3.3.1 Hypergraph Ramsey Theorem
100(2)
3.3.2 Deducing Arnautov-Folkman-Rado-Sanders' Theorem
102(5)
3.3.3 Symmetric Hypergraph Theorem
107(5)
3.4 Infinite Graphs
112(3)
3.4.1 Canonical Ramsey Theorem
114(1)
3.5 Comparing Ramsey and van der Waerden Results
115(5)
3.6 Exercises
120(3)
4 Euclidean Ramsey Theory
123(20)
4.1 Polygons
123(9)
4.2 Chromatic Number of the Plane
132(4)
4.3 Four Color Map Theorem
136(4)
4.4 Exercises
140(3)
5 Other Approaches to Ramsey Theory
143(40)
5.1 Topological Approaches
143(12)
5.1.1 Proof of van der Waerden's Theorem
143(3)
5.1.2 Proof of the de Bruijn-Erdos Theorem
146(3)
5.1.3 Proof of Hindman's Theorem
149(6)
5.2 Ergodic Theory
155(7)
5.2.1 Furstenberg's Proof of Szemeredi's Theorem
158(4)
5.3 Stone-Cech Compactification
162(8)
5.3.1 Proof of Schur's and Hindman's Theorems
165(3)
5.3.2 Proof of the de Bruijn-Erdos Theorem
168(2)
5.4 Additive Combinatorics Methods
170(9)
5.4.1 The Circle Method: Infinitely Many 3-Term Arithmetic Progressions among the Primes
170(4)
5.4.1.1 Minor Arcs
174(1)
5.4.1.2 Major Arcs
175(4)
5.5 Exercises
179(4)
6 The Probabilistic Method
183(24)
6.1 Lower Bounds on Ramsey, van der Waerden, and Hales-Jewett Numbers
183(4)
6.2 Turan's Theorem
187(3)
6.3 Almost-surely van der Waerden and Ramsey Numbers
190(5)
6.4 Lovasz Local Lemma
195(9)
6.5 Exercises
204(3)
7 Applications
207(14)
7.1 Fermat's Last Theorem
207(3)
7.2 Encoding Information
210(5)
7.3 Data Mining
215(4)
7.4 Exercises
219(2)
Bibliography 221(16)
Index 237
Aaron Robertson received his Ph.D. in mathematics from Temple University under the guidance of his advisor Doron Zeilberger. Upon finishing his Ph.D. he started at Colgate University in upstate New York where he is currently Professor of Mathematics. He also serves as Associate Managing editor of the journal Integers. After a brief detour into the world of permutation patterns, he has focused most of his research on Ramsey theory.
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