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More About This Textbook
Overview
About the Author:
Bela Bollobas is a Senior Research Fellow at Trinity College, Cambridge and holds the jabie Hardin Chair of Excellence in Combinatorics at the University of Memphis
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"The Author considers these problems to be the type that two mathematical friends would pose to each other and discuss over a cup of coffee in a lounge. I agree with that premise, they are not too hard and there is a proof that is relatively easy to discover and even easier to understand. These problems satisfy all of the requirements for a good problem..."Journal of Recreational Mathematics
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Table of Contents
Preface xiii
The Problems 1
The Hints 36
The Solutions 45
The Lion and The Christian 45
Integer Sequences: Erdos Problems for Epsilons 48
Points on a Circle 50
Partitions into Closed Sets 52
Triangles and Squares 53
Polygons and Rectangles 55
African Rally 56
Fixing Convex Domains 58
Nested Subsets 61
Almost Disjoint Subsets 63
Loaded Dice 64
An Unexpected Inequality 65
Colouring Lines: the ErdosSelfridge Theorem 66
Independent Sets 68
Expansion into Sums 2i 3j 69
A Tennis Match 70
A Triangle Inequality: Another Erdos Problem for Epsilons 71
Planar Domains of Diameter 1 73
Orienting Graphs 74
A Simple Clock 75
Neighbours in a Matrix 76
Separately Continuous Functions 77
Boundary Cubes 78
Lozenge Tilings 79
A Continuum Independent Set 83
Separating Families of Sets 84
Bipartite Covers of Complete Graphs 86
Convexity and Intersecting Simplices: the Theorems of Radon and Caratheodory 88
Intersecting Convex Sets: Helly's Theorem 90
Judicious Partitions of Points 92
Further Lozenge Tilings 93
Two Squares in a Square 95
Lines Through Points: the SylvesterGallai Theorem 98
The Spread of Infection on a Square Grid 104
The Spread of Infection in a ddimensional Box 106
Sums of Integers: an Easy Erdos Problem for Epsilons 110
Normal Numbers: the Champernowne Number 111
Random Walks on Graphs 113
Simple Tilings of Rectangles 114
Ltilings 116
Antipodal Points and Maps: Borsuk's Theorem 117
Bodies of Diameter 1: Borsuk's Problem 120
Equilateral Triangles: Napoleon's Theorem 124
Trisectors of Angles: Morley's Theorem 126
Connected Subgraphs 129
Subtrees of an Infinite Tree 133
Twodistance Sets 134
Gossiping Dons 136
Exact Covers: the de BruijnErdos Theorem 140
Constant Intersections: an Extension of the de BruijnErdos Theorem 142
Bell Numbers 144
Circles Touching a Square 147
Gambling 149
Complex Sequences 151
Partitions of Integers 153
Emptying Glasses 157
Distances in Planar Sets 159
Monic Polynomials 161
Odd Clubs 163
A Politically Correct Town 164
Lattice Paths 165
Triangulations of Polygons 168
A Converse of Cauchy's Inequality: Zagier's Inequality 169
Squares Touching a Square 170
Infection with Three Neighbours 171
The Spread of Infection on a Torus 173
Dominating Sequences 174
Sums of Reciprocals 175
Absentminded Passengers 176
Airline Luggage 177
Intersecting Sets: the ErdosKoRado Theorem 179
Sperner Families: the MYBL Inequality 180
Five Points in Space 183
Triads 184
Colouring Complete Graphs 186
Symmetric Convex Domains: a Theorem of Besicovitch 187
Independent Random Variables 190
Triangles Touching a Triangle 192
Even and Odd Graphs 193
Packing Squares: the MoonMoser Theorem 194
Filling a Matrix 197
An Inequality Concerning Triangles: the ErdosMordell Theorem 199
Perfect Difference Sets 203
Difference Bases 205
Satisfied Cricketers: the HardyLittlewood Maximal Theorem 208
Random Words 212
Crossing a Chess Board 214
Powers of Paths and Cycles 216
Powers of Oriented Cycles 217
Perfect Trees 218
Circular sequences 220
Infinite Sets with Integral Distances 222
Finite Sets with Integral Distances 223
Cubefree Words: Thue's Theorem 224
Squarefree Words: the ThueMorse Theorem 226
Addition of Residue Classes: the CauchyDavenport Theorem 229
Sums Congruent to Zero: the ErdosGinzburgZiv Theorem 232
Subwords of Distinct Words 237
Prime Factors of Sums 238
Catalan Numbers 240
Permutations without Long Decreasing Subsequences 242
Random Intervals: a Theorem of Justicz, Scheinerman and Winkler 244
Sums of Convex Bodies: the BrunnMinkowski Inequality 246
CrossIntersecting Families: Bollobas's Lemma 248
Saturated Hypergraphs 252
The Norm of Averages: Hardy's Inequality 253
The Average of Geometric Means: Carleman's Inequality 257
Triangulating Squares 259
Strongly Separating Families 262
Strongly Separating Systems of Pairs of Sets 263
The Maximum EdgeBoundary of a Downset 265
Partitioning a Subset of the Cube 267
Weakly Crossintersecting Pairs: Frankl's Theorem 269
Even Sets with Even Intersections 271
Sets with Even Intersections 273
Even Clubs 275
Covering the Sphere 276
The Kneser Graph: Lovasz's Theorem 277
Partitions into Bricks 279
Drawing Dense Graphs 280
Unit Distances: Szekely's Theorem 282
PointLine Incidences 284
Geometric Graphs without Parallel Edges 285
Shortest Tours 288
Density of Integers 291
Black and White Sheep: Kirchberger's Theorem 293
Chords of Convex Bodies 294
Neighourly Polyhedra 296
Neighbourly Simplices: Perles' Theorem 299
The Rank of a Matrix 301
Modular Intersecting kuniform Set Systems: a Theorem of Frankl and Wilson 303
Families without Orthogonal Vectors 306
A Counterexample to Borsuk's Conjecture: the KahnKalai Theorem 308
Periodic Sequences 311
Periodic Words: the FineWilf Theorem 313
Points on a Hemisphere: Wendel's Theorem 315
Planar and Spherical Triangles 318
Hobnails: Hadziivanov's theorem 319
A Probabilistic Inequality 321
Cube Slicing 322
Measures on [0, 1]: the HobbyRice Theorem 324
Cutting a Necklace 326
The Norm of an Operator: the RieszThorin Interpolation Theorem 328
Uniform Covers 332
Projections of Bodies 333
BTBT: the Box Theorem of Bollobas and Thomason 335
Intersecting Uniform Set Systems: the RayChaudhuriWilson Inequality 337
Intersecting Set Systems: the FranklWilson Inequality 340
Maps from S[superscript n] 342
Closed Covers of S[superscript n]: Hopf's Theorem 344
Spherical Pairs 345
Realizing Distances 345
A Closed Cover of S[superscript 2] 348
A Friendly Party: the Friendship Theorem of Erdos, Renyi and Sos 349
Polarities in Projective Planes 352
Permutations of Vectors: Steinitz's Theorem 353
The PointLine Game 356