Softcover ISBN:  9780821852811 
Product Code:  STML/56 
List Price:  $33.00 
Individual Price:  $26.40 
Electronic ISBN:  9781470416393 
Product Code:  STML/56.E 
List Price:  $31.00 
Individual Price:  $24.80 

Book DetailsStudent Mathematical LibraryVolume: 56; 2011; 150 ppMSC: Primary 05; 11; 42; 51;
The Erdős problem asks, What is the smallest possible number of distinct distances between points of a large finite subset of the Euclidean space in dimensions two and higher? The main goal of this book is to introduce the reader to the techniques, ideas, and consequences related to the Erdős problem. The authors introduce these concepts in a concrete and elementary way that allows a wide audience—from motivated high school students interested in mathematics to graduate students specializing in combinatorics and geometry—to absorb the content and appreciate its farreaching implications. In the process, the reader is familiarized with a wide range of techniques from several areas of mathematics and can appreciate the power of the resulting symbiosis.
The book is heavily problem oriented, following the authors' firm belief that most of the learning in mathematics is done by working through the exercises. Many of these problems are recently published results by mathematicians working in the area. The order of the exercises is designed both to reinforce the material presented in the text and, equally importantly, to entice the reader to leave all worldly concerns behind and launch head first into the multifaceted and rewarding world of Erdős combinatorics.ReadershipUndergraduates, graduate students, and research mathematicians interested in geometric combinatorics and various topics in general combinatorics.

Table of Contents

Chapters

Introduction

Chapter 1. The $\sqrt {n}$ theory

Chapter 2. The $n^{2/3}$ theory

Chapter 3. The CauchySchwarz inequality

Chapter 4. Graph theory and incidences

Chapter 5. The $n^{4/5}$ theory

Chapter 6. The $n^{6/7}$ theory

Chapter 7. Beyond $n^{6/7}$

Chapter 8. Information theory

Chapter 9. Dot products

Chapter 10. Vector spaces over finite fields

Chapter 11. Distances in vector spaces over finite fields

Chapter 12. Applications of the Erdős distance problem

Appendix A. Hyperbolas in the plane

Appendix B. Basic probability theory

Appendix C. Jensen’s inequality


Additional Material

Reviews

The authors do an excellent job in bringing together the main techniques and results connected to the Erdős distance problem ... this is a useful book for the reader with sufficient mathematical experience who wishes to learn the principal techniques and results in the Erdős distance problem and related areas.
Mathematical Reviews 
This book...achieves the remarkable feat of providing an extremely accessible treatment of a classic family of research problems. ...The book can be used for a reading course taken by an undergraduate student (parts of the book are accessible for talented high school students as well), or as introductory material for a graduate student who plans to investigate this area further...Highly recommended.
M. Bona, Choice


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The Erdős problem asks, What is the smallest possible number of distinct distances between points of a large finite subset of the Euclidean space in dimensions two and higher? The main goal of this book is to introduce the reader to the techniques, ideas, and consequences related to the Erdős problem. The authors introduce these concepts in a concrete and elementary way that allows a wide audience—from motivated high school students interested in mathematics to graduate students specializing in combinatorics and geometry—to absorb the content and appreciate its farreaching implications. In the process, the reader is familiarized with a wide range of techniques from several areas of mathematics and can appreciate the power of the resulting symbiosis.
The book is heavily problem oriented, following the authors' firm belief that most of the learning in mathematics is done by working through the exercises. Many of these problems are recently published results by mathematicians working in the area. The order of the exercises is designed both to reinforce the material presented in the text and, equally importantly, to entice the reader to leave all worldly concerns behind and launch head first into the multifaceted and rewarding world of Erdős combinatorics.
Undergraduates, graduate students, and research mathematicians interested in geometric combinatorics and various topics in general combinatorics.

Chapters

Introduction

Chapter 1. The $\sqrt {n}$ theory

Chapter 2. The $n^{2/3}$ theory

Chapter 3. The CauchySchwarz inequality

Chapter 4. Graph theory and incidences

Chapter 5. The $n^{4/5}$ theory

Chapter 6. The $n^{6/7}$ theory

Chapter 7. Beyond $n^{6/7}$

Chapter 8. Information theory

Chapter 9. Dot products

Chapter 10. Vector spaces over finite fields

Chapter 11. Distances in vector spaces over finite fields

Chapter 12. Applications of the Erdős distance problem

Appendix A. Hyperbolas in the plane

Appendix B. Basic probability theory

Appendix C. Jensen’s inequality

The authors do an excellent job in bringing together the main techniques and results connected to the Erdős distance problem ... this is a useful book for the reader with sufficient mathematical experience who wishes to learn the principal techniques and results in the Erdős distance problem and related areas.
Mathematical Reviews 
This book...achieves the remarkable feat of providing an extremely accessible treatment of a classic family of research problems. ...The book can be used for a reading course taken by an undergraduate student (parts of the book are accessible for talented high school students as well), or as introductory material for a graduate student who plans to investigate this area further...Highly recommended.
M. Bona, Choice