Introduction

Many theorems in mathematics say, in one way or another, that it

is very diﬃcult to arrange mathematical objects in such a way that

they do not exhibit some interesting structure. The objects in the

Erd˝ os distance problem are points, and the structure we are curious

about involves distances between points. We can loosely formulate the

main question of this book as follows: How many distinct distances

are determined by a finite set of points?

1. A sketch of our problem

In the case that there is only one point, we have but one distance,

zero. It might seem odd to count zero as a distance, but it will make

things easier later on if we just assume that it is. In the case of two

points, our job is pretty easy again. We have the distance between the

two points, and again, zero. However, if we consider the case of three

points in the plane, it begins to get interesting. Three points arranged

as the vertices of an equilateral triangle are the same distance from

one another, so there is only one nonzero distance, making two total.

If they are the vertices of an isosceles triangle, we have one distance

repeated, leaving three distinct distances total. Of course, there are

any number of ways for three points to determine four distances.

These phenomena increase in complexity and frequency as we consider

more and more points. In fact, there is no configuration of four points

1

http://dx.doi.org/10.1090/stml/056/01