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Index acyclic directed graph, 196 addition theorems, 150 adjacency matrix, 248 adjacent vertices, 66 antichain, 179 arithmetic mean, 119 art gallery theorem, 232 average degree, 76 average number of divisors, 164 Bernoulli numbers, 48, 152 Bertrand’s postulate, 7 bijection, 103, 207 Binet–Cauchy formula, 197, 203 binomial coefficient, 13 bipartite graph, 67, 223 birthday paradox, 185 Bolyai–Gerwien Theorem, 53 Borsuk’s conjecture, 95 Borsuk–Ulam theorem, 252 Bricard’s condition, 57 Brouwer’s fixed point theorem, 169 Buffon’s needle problem, 155 Calkin–Wilf tree, 105 Cantor–Bernstein theorem, 110 capacity, 242 cardinal number, 103 cardinality, 103, 115 Cauchy’s arm lemma, 82 Cauchy’s minimum principle, 127 Cauchy’s rigidity theorem, 81 Cauchy–Schwarz inequality, 119 Cayley’s formula, 201 center, 31 centralizer, 31 centrally symmetric, 61 chain, 179 channel, 241 Chebyshev polynomials, 144 Chebyshev’s theorem, 140 chromatic number, 221, 251 class formula, 32 clique, 67, 235, 242 clique number, 237 2-colorable set system, 261 combinatorially equivalent, 61 comparison of coefficients, 47 complete bipartite graph, 66 complete graph, 66 complex polynomial, 139 components of a graph, 67 cone lemma, 56 confusion graph, 241 congruent, 61 connected, 67 connected components, 67 continuum, 109 continuum hypothesis, 112 convex polytope, 59 convex vertex, 232 cosine polynomial, 143 countable, 103 coupon collector’s problem, 186 critical family, 182 crossing lemma, 267 crossing number, 266 cube, 60 cycle, 67 C 4 -condition, 257 C 4 -free graph, 166 degree, 76 dense, 114 determinants, 195 dihedral angle, 57 dimension, 109 dimension of a graph, 162 Dinitz problem, 221 directed graph, 223
272 Index division ring, 31 double counting, 164 dual graph, 75, 227 edge of a graph, 66 edge of a polyhedron, 60 elementary polygon, 79 equal size, 103 equicomplementability, 54 equicomplementable polyhedra, 53 equidecomposability, 54 equidecomposable polyhedra, 53 Erdős–Ko–Rado theorem, 180 Euler’s criterion, 24 Euler’s function, 28 Euler’s polyhedron formula, 75 Euler’s series, 43 even function, 152 expectation, 94 face, 60, 75 facet, 60 Fermat number, 3 finite field, 31 finite fields, 28 finite set system, 179 forest, 67 formal power series, 207 four-color theorem, 227 friendship theorem, 257 fundamental theorem of algebra, 127 Gale’s theorem, 253 Gauss lemma, 25 Gauss sum, 27 general position, 253 geometric mean, 119 Gessel–Viennot lemma, 195 girth, 263 golden section, 245 graph, 66 graph coloring, 227 graph of a polytope, 60 harmonic mean, 119 harmonic number, 10 Herglotz trick, 149 Hilbert’s third problem, 53 hyper-binary representation, 106 incidence matrix, 64, 164 incident, 66 indegree, 223 independence number, 241, 251, 264 independent set, 67, 221 induced subgraph, 67, 222 inequalities, 119 infinite products, 207 initial ordinal number, 116 intersecting family, 180, 251 involution, 20 irrational numbers, 35 isomorphic graphs, 67 Jacobi determinants, 45 kernel, 223 Kneser graph, 251 Kneser’s conjecture, 252 labeled tree, 201 Lagrange’s theorem, 4 Latin rectangle, 214 Latin square, 213, 221 lattice, 79 lattice basis, 80 lattice paths, 195 lattice points, 26 law of quadratic reciprocity, 24 Legendre symbol, 23 Legendre’s theorem, 8 lexicographically smallest solution, 56 line graph, 226 linear extension, 176 linearity of expectation, 94, 156 list chromatic number, 222 list coloring, 222, 228 Littlewood–Offord problem, 145 loop, 66 Lovász’ theorem, 247 Lovász umbrella, 244 Lyusternik–Shnirel’man theorem, 252 Markov’s inequality, 94 marriage theorem, 182 matching, 224 matrix of rank 1, 96
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Martin Aigner Günter M. Ziegler Pr
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Prof. Dr. Martin Aigner FB Mathemat
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VI Preface to the Fourth Edition Wh
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VIII Table of Contents 21. A theore
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Six proofs of the infinity of prime
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Six proofs of the infinity of prime
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Bertrand’s postulate Chapter 2 We
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Bertrand’s postulate 9 times. Her
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Bertrand’s postulate 11 by compar
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Binomial coefficients are (almost)
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Binomial coefficients are (almost)
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Representing numbers as sums of two
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The law of quadratic reciprocity Ch
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The law of quadratic reciprocity 25
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Every finite division ring is a fie
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Every finite division ring is a fie
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Some irrational numbers Chapter 7
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Some irrational numbers 37 and this
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Some irrational numbers 39 F(x) may
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Some irrational numbers 41 Now assu
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44 Three times π 2 /6 v 1 1 2 y v
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46 Three times π 2 /6 For m = 1, 2
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48 Three times π 2 /6 1 f(t) = 1 t
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50 Three times π 2 /6 (3) It has b
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Hilbert’s third problem: decompos
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64 Lines in the plane, and decompos
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66 Lines in the plane, and decompos
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The slope problem Chapter 11 Try fo
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The slope problem 71 • Every move
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The slope problem 73 For the second
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76 Three applications of Euler’s
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82 Cauchy’s rigidity theorem Q: Q
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84 Cauchy’s rigidity theorem A be
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86 Touching simplices l l l ′ Pr
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88 Touching simplices For our examp
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90 Every large point set has an obt
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96 Borsuk’s conjecture Theorem. L
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98 Borsuk’s conjecture On the oth
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100 Borsuk’s conjecture To obtain
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Sets, functions, and the continuum
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In praise of inequalities Chapter 1
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In praise of inequalities 121 where
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In praise of inequalities 123 Proo
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In praise of inequalities 125 Seco
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128 The fundamental theorem of alge
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One square and an odd number of tri
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A theorem of Pólya on polynomials
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On a lemma of Littlewood and Offord
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On a lemma of Littlewood and Offord
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Cotangent and the Herglotz trick Ch
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Cotangent and the Herglotz trick 15
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Cotangent and the Herglotz trick 15
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Buffon’s needle problem Chapter 2
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Buffon’s needle problem 157 The c
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Combinatorics 25 Pigeon-hole and do
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162 Pigeon-hole and double counting
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Tiling rectangles Chapter 26 Some m
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Tiling rectangles 175 Third proof.
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Tiling rectangles 177 References [1
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180 Three famous theorems on finite
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182 Three famous theorems on finite
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Shuffling cards Chapter 28 How ofte
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Shuffling cards 187 Indeed, if A i
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Shuffling cards 189 Let T be the nu
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Shuffling cards 191 Theorem 1. Let
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Shuffling cards 193 Proof. (1) We
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Lattice paths and determinants Chap
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Lattice paths and determinants 197
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Lattice paths and determinants 199
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Cayley’s formula for the number o
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Identities versus bijections Chapte
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Identities versus bijections 209 Pr
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Identities versus bijections 211 As
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Completing Latin squares Chapter 32
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Completing Latin squares 215 Proof
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Completing Latin squares 217 be dis
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Graph Theory 33 The Dinitz problem
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Page 237 and 238: 228 Five-coloring plane graphs From
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Page 241 and 242: 232 How to guard a museum The follo
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Page 278 and 279: Proofs from THE BOOK 269
Page 282 and 283: Index 273 matrix-tree theorem, 203
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