# The Probabilistic Method, 4th Edition

# The Probabilistic Method, 4th Edition

ISBN: 978-1-119-06195-3

Jan 2016

400 pages

## Description

**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.

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PREFACE xiii

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 𝜖-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

"This is an ideal textbook for upper-undergraduate and graduate-level students majoring in mathematics, computer science, operations research, and statistics." (*Springer Nature*, 2016)