The Erdos 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 Erdos 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 far-reaching 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 Erdos combinatorics.

Introduction
Chapter 1. The n theory
Chapter 2. The n2/3 theory
Chapter 3. The Cauchy-Schwarz inequality
Chapter 4. Graph theory and incidences
Chapter 5. The n4/5 theory
Chapter 6. The n6/7 theory
Chapter 7. Beyond n6/7
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