**Student Mathematical Library**

Volume: 87;
2018;
207 pp;
Softcover

MSC: Primary 05; 03;

Print ISBN: 978-1-4704-4290-3

Product Code: STML/87

List Price: $52.00

Individual Price: $41.60

MAA Member Price: $41.60

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**Electronic ISBN: 978-1-4704-4994-0
Product Code: STML/87.E**

List Price: $52.00

Individual Price: $41.60

MAA Member Price: $41.60

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#### Supplemental Materials

# An Introduction to Ramsey Theory: Fast Functions, Infinity, and Metamathematics

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*Matthew Katz; Jan Reimann*

This book takes the reader on a journey through Ramsey theory, from
graph theory and combinatorics to set theory to logic and
metamathematics. Written in an informal style with few requisites, it
develops two basic principles of Ramsey theory: many combinatorial
properties persist under partitions, but to witness this persistence,
one has to start with very large objects. The interplay between those
two principles not only produces beautiful theorems but also touches
the very foundations of mathematics. In the course of this book, the
reader will learn about both aspects. Among the topics explored are
Ramsey's theorem for graphs and hypergraphs, van der Waerden's theorem
on arithmetic progressions, infinite ordinals and cardinals, fast
growing functions, logic and provability, Gödel incompleteness, and
the Paris-Harrington theorem.

Quoting from the book, “There seems to be a murky abyss
lurking at the bottom of mathematics. While in many ways we cannot
hope to reach solid ground, mathematicians have built impressive
ladders that let us explore the depths of this abyss and marvel at the
limits and at the power of mathematical reasoning at the same
time. Ramsey theory is one of those ladders.”

This book is published in cooperation with Mathematics Advanced Study Semesters

#### Readership

Undergraduate and graduate students and researchers interested in combinatorics and mathematical logic.

#### Reviews & Endorsements

This little book is a gem. It is advertised as having minimal prerequisites. Such statements are often made but less often accurate, but the claim seems entirely truthful here. The authors have taken pains to develop, as necessary, what background material is used in the book...and have written this book in a clear, conversational tone that is certain to engage student interest.

-- Mark Hunacek, MAA Reviews

#### Table of Contents

# Table of Contents

## An Introduction to Ramsey Theory: Fast Functions, Infinity, and Metamathematics

- Cover Cover11
- Title page iii4
- Foreword: MASS at Penn State University vii8
- Preface ix10
- Chapter 1. Graph Ramsey theory 116
- Chapter 2. Infinite Ramsey theory 4156
- 2.1. The infinite Ramsey theorem 4156
- 2.2. König’s lemma and compactness 4358
- 2.3. Some topology 5065
- 2.4. Ordinals, well-orderings, and the axiom of choice 5570
- 2.5. Cardinality and cardinal numbers 6479
- 2.6. Ramsey theorems for uncountable cardinals 7085
- 2.7. Large cardinals and Ramsey cardinals 8095

- Chapter 3. Growth of Ramsey functions 85100
- Chapter 4. Metamathematics 129144
- 4.1. Proof and truth 129144
- 4.2. Non-standard models of Peano arithmetic 145160
- 4.3. Ramsey theory in Peano arithmetic 152167
- 4.4. Incompleteness 159174
- 4.5. Indiscernibles 171186
- 4.6. Diagonal indiscernibles via Ramsey theory 182197
- 4.7. The Paris-Harrington theorem 188203
- 4.8. More incompleteness 193208

- Bibliography 199214
- Notation 203218
- Index 205220
- Back Cover Back Cover1224