The Probabilistic Method
Wiley-Interscience Series in Discrete Mathematics and Optimization
4. Edition March 2016
400 Pages, Hardcover
Wiley & Sons Ltd
Praise for the Third Edition
"Researchers of any kind of extremal combinatorics or theoretical computer science will welcome the new edition of this book." - MAA Reviews
Maintaining a standard of excellence that establishes The Probabilistic Method as the leading reference on probabilistic methods in combinatorics, the Fourth Edition continues to feature a clear writing style, illustrative examples, and illuminating exercises. The new edition includes numerous updates to reflect the most recent developments and advances in discrete mathematics and the connections to other areas in mathematics, theoretical computer science, and statistical physics.
Emphasizing the methodology and techniques that enable problem-solving, The Probabilistic Method, Fourth Edition begins with a description of tools applied to probabilistic arguments, including basic techniques that use expectation and variance as well as the more advanced applications of martingales and correlation inequalities. The authors explore where probabilistic techniques have been applied successfully and also examine topical coverage such as discrepancy and random graphs, circuit complexity, computational geometry, and derandomization of randomized algorithms. Written by two well-known authorities in the field, the Fourth Edition features:
* Additional exercises throughout with hints and solutions to select problems in an appendix to help readers obtain a deeper understanding of the best methods and techniques
* New coverage on topics such as the Local Lemma, Six Standard Deviations result in Discrepancy Theory, Property B, and graph limits
* Updated sections to reflect major developments on the newest topics, discussions of the hypergraph container method, and many new references and improved results
The Probabilistic Method, Fourth Edition is an ideal textbook for upper-undergraduate and graduate-level students majoring in mathematics, computer science, operations research, and statistics. The Fourth Edition is also an excellent reference for researchers and combinatorists who use probabilistic methods, discrete mathematics, and number theory.
Noga Alon, PhD, is Baumritter Professor of Mathematics and Computer Science at Tel Aviv University. He is a member of the Israel National Academy of Sciences and Academia Europaea. A coeditor of the journal Random Structures and Algorithms, Dr. Alon is the recipient of the Polya Prize, The Gödel Prize, The Israel Prize, and the EMET Prize.
Joel H. Spencer, PhD, is Professor of Mathematics and Computer Science at the Courant Institute of New York University. He is the cofounder and coeditor of the journal Random Structures and Algorithms and is a Sloane Foundation Fellow. Dr. Spencer has written more than 200 published articles and is the coauthor of Ramsey Theory, Second Edition, also published by Wiley.
ACKNOWLEDGMENTS xv
PART I METHODS 1
1 The Basic Method 3
1.1 The Probabilistic Method, 3
1.2 Graph Theory, 5
1.3 Combinatorics, 9
1.4 Combinatorial Number Theory, 11
1.5 Disjoint Pairs, 12
1.6 Independent Sets and List Coloring, 13
1.7 Exercises, 16
The Erd?os-Ko-Rado Theorem, 18
2 Linearity of Expectation 19
2.1 Basics, 19
2.2 Splitting Graphs, 20
2.3 Two Quickies, 22
2.4 Balancing Vectors, 23
2.5 Unbalancing Lights, 25
2.6 Without Coin Flips, 26
2.7 Exercises, 27
Brégman's Theorem, 29
3 Alterations 31
3.1 Ramsey Numbers, 31
3.2 Independent Sets, 33
3.3 Combinatorial Geometry, 34
3.4 Packing, 35
3.5 Greedy Coloring, 36
3.6 Continuous Time, 38
3.7 Exercises, 41
High Girth and High Chromatic Number, 43
4 The Second Moment 45
4.1 Basics, 45
4.2 Number Theory, 46
4.3 More Basics, 49
4.4 Random Graphs, 51
4.5 Clique Number, 55
4.6 Distinct Sums, 57
4.7 The Rödl nibble, 58
4.8 Exercises, 64
Hamiltonian Paths, 65
5 The Local Lemma 69
5.1 The Lemma, 69
5.2 Property B and Multicolored Sets of Real Numbers, 72
5.3 Lower Bounds for Ramsey Numbers, 73
5.4 A Geometric Result, 75
5.5 The Linear Arboricity of Graphs, 76
5.6 Latin Transversals, 80
5.7 Moser's Fix-It Algorithm, 81
5.8 Exercises, 87
Directed Cycles, 88
6 Correlation Inequalities 89
6.1 The Four Functions Theorem of Ahlswede and Daykin, 90
6.2 The FKG Inequality, 93
6.3 Monotone Properties, 94
6.4 Linear Extensions of Partially Ordered Sets, 97
6.5 Exercises, 99
Turán's Theorem, 100
7 Martingales and Tight Concentration 103
7.1 Definitions, 103
7.2 Large Deviations, 105
7.3 Chromatic Number, 107
7.4 Two General Settings, 109
7.5 Four Illustrations, 113
7.6 Talagrand's Inequality, 116
7.7 Applications of Talagrand's Inequality, 119
7.8 Kim-Vu Polynomial Concentration, 121
7.9 Exercises, 123
Weierstrass Approximation Theorem, 124
8 The Poisson Paradigm 127
8.1 The Janson Inequalities, 127
8.2 The Proofs, 129
8.3 Brun's Sieve, 132
8.4 Large Deviations, 135
8.5 Counting Extensions, 137
8.6 Counting Representations, 139
8.7 Further Inequalities, 142
8.8 Exercises, 143
Local Coloring, 144
9 Quasirandomness 147
9.1 The Quadratic Residue Tournaments, 148
9.2 Eigenvalues and Expanders, 151
9.3 Quasirandom Graphs, 157
9.4 Szemerédi's Regularity Lemma, 165
9.5 Graphons, 170
9.6 Exercises, 172
Random Walks, 174
PART II TOPICS 177
10 Random Graphs 179
10.1 Subgraphs, 180
10.2 Clique Number, 183
10.3 Chromatic Number, 184
10.4 Zero-One Laws, 186
10.5 Exercises, 193
Counting Subgraphs, 195
11 The Erd?os-Rényi Phase Transition 197
11.1 An Overview, 197
11.2 Three Processes, 199
11.3 The Galton-Watson Branching Process, 201
11.4 Analysis of the Poisson Branching Process, 202
11.5 The Graph Branching Model, 204
11.6 The Graph and Poisson Processes Compared, 205
11.7 The Parametrization Explained, 207
11.8 The Subcritical Regions, 208
11.9 The Supercritical Regimes, 209
11.10 The Critical Window, 212
11.11 Analogies to Classical Percolation Theory, 214
11.12 Exercises, 219
Long paths in the supercritical regime, 220
12 Circuit Complexity 223
12.1 Preliminaries, 223
12.2 Random Restrictions and Bounded-Depth Circuits, 225
12.3 More on Bounded-Depth Circuits, 229
12.4 Monotone Circuits, 232
12.5 Formulae, 235
12.6 Exercises, 236
Maximal Antichains, 237
13 Discrepancy 239
13.1 Basics, 239
13.2 Six Standard Deviations Suffice, 241
13.3 Linear and Hereditary Discrepancy, 245
13.4 Lower Bounds, 248
13.5 The Beck-Fiala Theorem, 250
13.6 Exercises, 251
Unbalancing Lights, 253
14 Geometry 255
14.1 The Greatest Angle Among Points in Euclidean Spaces, 256
14.2 Empty Triangles Determined by Points in the Plane, 257
14.3 Geometrical Realizations of Sign Matrices, 259
14.4 epsilon-Nets and VC-Dimensions of Range Spaces, 261
14.5 Dual Shatter Functions and Discrepancy, 266
14.6 Exercises, 269
Efficient Packing, 270
15 Codes, Games, and Entropy 273
15.1 Codes, 273
15.2 Liar Game, 276
15.3 Tenure Game, 278
15.4 Balancing Vector Game, 279
15.5 Nonadaptive Algorithms, 281
15.6 Half Liar Game, 282
15.7 Entropy, 284
15.8 Exercises, 289
An Extremal Graph, 291
16 Derandomization 293
16.1 The Method of Conditional Probabilities, 293
16.2 d-Wise Independent Random Variables in Small Sample Spaces, 297
16.3 Exercises, 302
Crossing Numbers, Incidences, Sums and Products, 303
17 Graph Property Testing 307
17.1 Property Testing, 307
17.2 Testing Colorability, 308
17.3 Testing Triangle-Freeness, 312
17.4 Characterizing the Testable Graph Properties, 314
17.5 Exercises, 316
Turán Numbers and Dependent Random Choice, 317
Appendix A Bounding of Large Deviations 321
A.1 Chernoff Bounds, 321
A.2 Lower Bounds, 330
A.3 Exercises, 334
Triangle-Free Graphs Have Large Independence Numbers, 336
Appendix B Paul Erd?os 339
B.1 Papers, 339
B.2 Conjectures, 341
B.3 On Erd?os, 342
B.4 Uncle Paul, 343
The Rich Get Richer, 346
Appendix C Hints to Selected Exercises 349
REFERENCES 355
AUTHOR INDEX 367
SUBJECT INDEX 371
Joel H. Spencer, PhD, is Professor of Mathematics and Computer Science at the Courant Institute of New York University. He is the cofounder and coeditor of the journal Random Structures and Algorithms and is a Sloane Foundation Fellow. Dr. Spencer has written more than 200 published articles and is the coauthor of Ramsey Theory, Second Edition, also published by Wiley.
