Proofs from the books

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Martin Aigner
Giinter M. Ziegler

Proofs from
Third Edition
With 250 Figures
Including Illustrations
by Karl H. Hofmann



Paul Erd˝ s liked to talk about The Book, in which God maintains the perfect
proofs for mathematical theorems, following the dictum of G. H. Hardy that
there is no permanent place for ugly mathematics. Erd˝ s also said that you
need not believe in God but, as a mathematician, you should believe in
The Book. A few years ago, we suggested to him to write up a first (and
very modest) approximation to The Book. He was enthusiastic about the
idea and, characteristically, went to work immediately, filling page after
page with his suggestions. Our book was supposed to appear in March
1998 as a present to Erd˝ s’ 85th birthday. With Paul’s unfortunate death
in the summer of 1997, he is not listed as a co-author. Instead this book is
dedicated to his memory.
We have no definition or characterization of what constitutes a proof from
The Book: all we offer here is the examples that we have selected, hoping that our readers will share our enthusiasm about brilliant ideas, clever
insights and wonderful observations. We also hope that our readers will
enjoy this despite the imperfections of our exposition. The selection is to a
great extent influenced by Paul Erd˝ s himself. A large number of the topics
were suggested by him, and many of the proofs trace directly back to him,
or were initiated by his supreme insight in asking the right question or in
making the right conjecture. So to a large extent this book reflects the views
of Paul Erd˝ s as to what should be considered a proof from The Book.
A limiting factor for our selection of topics was that everything in this book
is supposed to be accessible to readers whose backgrounds include only
a modest amount of technique from undergraduate mathematics. A little
linear algebra, some basic analysis and number theory, and a healthy dollop
of elementary concepts and reasonings from discrete mathematics should
be sufficient to understand and enjoy everything in this book.
We are extremely grateful to the many people who helped and supported
us with this project — among them the students of a seminar where we
discussed a preliminary version, to Benno Artmann, Stephan Brandt, Stefan
Felsner, Eli Goodman, Torsten Heldmann, and Hans Mielke. We thank
Margrit Barrett, Christian Bressler, Ewgenij Gawrilow, Elke Pose, and J¨ rg
Rambau for their technical help in composing this b...
Martin Aigner
Giinter M. Ziegler
Third Edition
Including Illustrations
Karl H. Hofmann
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