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-Čech 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.
Author(s): Aaron Robertson
Series: Discrete Mathematics and its Applications
Edition: 1
Publisher: CRC Press
Year: 2021
Language: English
Pages: 255
Tags: Ramsey Theory
Cover
Half Title
Series Page
Title Page
Copyright Page
Dedication
Contents
Preface
Symbols
1. Introduction
1.1. What is Ramsey Theory?
1.2. Notations and Conventions
1.3. Prerequisites
1.3.1. Combinatorics
1.3.2. Analysis
1.3.3. Probability
1.3.4. Algebra
1.3.5. Topology
1.3.6. Statistics
1.3.7. Practice
1.4. Compactness Principle
1.5. Set Theoretic Considerations
1.6. Exercises
2. Integer Ramsey Theory
2.1. Van der Waerden's Theorem
2.1.1. Hilbert's Cube Lemma
2.1.2. Deuber's Theorem
2.2. Equations
2.2.1. Schur's Theorem
2.2.2. Rado's Theorem
2.2.2.1. Some Rado Numbers
2.2.3. Nonlinear Equations
2.2.4. Algebraic Equations
2.3. Hales-Jewett Theorem
2.4. Finite Sums
2.4.1. Arnautov-Folkman-Rado-Sanders' Theorem
2.4.2. Hindman's Theorem
2.5. Density Results
2.5.1. Roth's Theorem
2.5.2. Szemeredi's Theorem
2.5.3. Density Hales-Jewett Theorem
2.6. Exercises
3. Graph Ramsey Theory
3.1. Complete Graphs
3.1.1. Deducing Schur's Theorem
3.2. Other Graphs
3.2.1. Some Graph Theory Concepts
3.2.2. Graph Ramsey Numbers
3.3. Hypergraphs
3.3.1. Hypergraph Ramsey Theorem
3.3.2. Deducing Arnautov-Folkman-Rado-Sanders' Theorem
3.3.3. Symmetric Hypergraph Theorem
3.4. Infinite Graphs
3.4.1. Canonical Ramsey Theorem
3.5. Comparing Ramsey and van der Waerden Results
3.6. Exercises
4. Euclidean Ramsey Theory
4.1. Polygons
4.2. Chromatic Number of the Plane
4.3. Four Color Map Theorem
4.4. Exercises
5. Other Approaches to Ramsey Theory
5.1. Topological Approaches
5.1.1. Proof of van der Waerden's Theorem
5.1.2. Proof of the de Bruijn-Erdos Theorem
5.1.3. Proof of Hindman's Theorem
5.2. Ergodic Theory
5.2.1. Furstenberg's Proof of Szemeredi's Theorem
5.3. Stone-Cech Compactification
5.3.1. Proof of Schur's and Hindman's Theorems
5.3.2. Proof of the de Bruijn-Erdos Theorem
5.4. Additive Combinatorics Methods
5.4.1. The Circle Method: Infinitely Many 3-term Arithmetic Progressions Among the Primes
5.4.1.1. Minor Arcs
5.4.1.2. Major Arcs
5.5. Exercises
6. The Probabilistic Method
6.1. Lower Bounds on Ramsey, van der Waerden, and Hales-Jewett Numbers
6.2. Turan's Theorem
6.3. Almost-surely van der Waerden and Ramsey Numbers
6.4. Lovasz Local Lemma
6.5. Exercises
7. Applications
7.1. Fermat's Last Theorem
7.2. Encoding Information
7.3. Data Mining
7.4. Exercises
Bibliography
Index
