There are several goals for this book. As the title indicates, we cer-
tainly hope to familiarize you with some of the major results in the
study of the Erd˝ os distance problem. This goal should be easily at-
tainable for most experienced mathematicians. However, if you are
not an experienced mathematician, we hope to guide you through
many advanced mathematical concepts along the way.
The book is based on the notes that were written for the summer
program on the problem, held at the University of Missouri, August
1–5, 2005. This was the second year of the program, and our plan
continued to be an introduction for motivated high school students
to accessible concepts of higher mathematics.
This book is designed to be enjoyed by readers at different levels
of mathematical experience. Keep in mind that some of the notes
and remarks are directed at graduate students and professionals in
the field. So, if you are relatively inexperienced, and a particular
comment or observation uses terminology1 that you are not familiar
with, you may want to skip past it or look up the definitions later.
On the other hand, if you are a more experienced mathematician, feel
free to skim the introductory portions to glean the necessary notation,
and move on to the more specific subject matter.
example of this is the mention of curvature in the first section of the
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