Preface Paul Erdos 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.Erdos 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 Erdos'85th birthday.With Paul's unfortunate death in the summer of 1996,he is not listed as a co-author.Instead this book is dedicated to his memory. Paul Erd6s 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,hop- ing 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 Erdos 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 Erdos as to what should be considered a proof from The Book “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,Michael Joswig, Elke Pose,and Jorg Rambau for their technical help in composing this book.We are in great debt to Tom Trotter who read the manuscript from first to last page,to Karl H.Hofmann for his wonderful drawings,and most of all to the late great Paul Erdos himself. Berlin,March 1998 Martin Aigner.Giinter M.ZieglerPreface Paul Erdos˝ Paul Erdos 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. Erdos 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 Erdos’ 85th birthday. With Paul’s unfortunate death ˝ in the summer of 1996, he is not listed as a co-author. Instead this book is dedicated to his memory. “The Book” 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, hop￾ing 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 Erdos 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 Erdos 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, Michael Joswig, Elke Pose, and Jörg Rambau for their technical help in composing this book. We are in great debt to Tom Trotter who read the manuscript from first to last page, to Karl H. Hofmann for his wonderful drawings, and most of all to the late great Paul Erdos himself. ˝ Berlin, March 1998 Martin Aigner · Günter M. Ziegler