The Probabilistic Method, Third Edition
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Praise for the Second Edition:

"Serious researchers in combinatorics or algorithm design will wish to read the book in its entirety...the book may also be enjoyed on a lighter level since the different chapters are largely independent and so it is possible to pick out gems in one's own area..."
Formal Aspects of Computing

This Third Edition of The Probabilistic Method reflects the most recent developments in the field while maintaining the standard of excellence that established this book as the leading reference on probabilistic methods in combinatorics. Maintaining its clear writing style, illustrative examples, and practical exercises, this new edition emphasizes methodology, enabling readers to use probabilistic techniques for solving problems in such fields as theoretical computer science, mathematics, and statistical physics.

The book begins with a description of tools applied in probabilistic arguments, including basic techniques that use expectation and variance as well as the more recent applications of martingales and correlation inequalities. Next, the authors examine where probabilistic techniques have been applied successfully, exploring such topics as discrepancy and random graphs, circuit complexity, computational geometry, and derandomization of randomized algorithms. Sections labeled "The Probabilistic Lens" offer additional insights into the application of the probabilistic approach, and the appendix has been updated to include methodologies for finding lower bounds for Large Deviations.

The Third Edition also features:

A new chapter on graph property testing, which is a current topic that incorporates combinatorial, probabilistic, and algorithmic techniques

An elementary approach using probabilistic techniques to the powerful Szemerédi Regularity Lemma and its applications

New sections devoted to percolation and liar games

A new chapter that provides a modern treatment of the Erdös-Rényi phase transition in the Random Graph Process

Written by two leading authorities in the field, The Probabilistic Method, Third Edition is an ideal reference for researchers in combinatorics and algorithm design who would like to better understand the use of probabilistic methods. The book's numerous exercises and examples also make it an excellent textbook for graduate-level courses in mathematics and computer science.


NOGA ALON, PhD, is Baumritter Professor of Mathematics and Computer Science at Tel Aviv University, Israel. A member of the Israel National Academy of Sciences, Dr. Alon has written over 400 published papers, mostly in the areas of combinatorics and theoretical computer science. He is the recipient of numerous honors in the field, including the Erdös Prize (1989), the Pólya Prize (2000), the Landau Prize (2005), and the Gödel Prize (2005).

JOEL H. SPENCER, PhD, is Professor of Mathematics and Computer Science at the Courant Institute of Mathematical Sciences at New York University and is the cofounder and coeditor of the journal Random Structures and Algorithms. Dr. Spencer has written over 150 published articles and is the coauthor of Ramsey Theory, Second Edition, also published by Wiley.






1. The Basic Method.

1.1 The Probabilistic Method.

1.2 Graph Theory.

1.3 Combinatorics.

1.4 Combinatorial Number Theory.

1.5 Disjoint Pairs.

1.6 Exercises.

The Probabilistic Lens: The Erd" osKoRado Theorem.

2. Linearity of Expectation.

2.1 Basics.

2.2 Splitting Graphs.

2.3 Two Quickies.

2.4 Balancing Vectors.

2.5 Unbalancing Lights.

2.6 Without Coin Flips.

2.7 Exercises.

The Probabilistic Lens: Brégman’s Theorem.

3. Alterations.

3.1 Ramsey Numbers.

3.2 Independent Sets.

3.3 Combinatorial Geometry.

3.4 Packing.

3.5 Recoloring.

3.6 Continuous Time.

3.7 Exercises.

The Probabilistic Lens: High Girth and High Chromatic Number.

4. The Second Moment.

4.1 Basics.

4.2 Number Theory.

4.3 More Basics.

4.4 Random Graphs.

4.5 Clique Number.

4.6 Distinct Sums.

4.7 The Rödl Nibble.

4.8 Exercises.

The Probabilistic Lens: Hamiltonian Paths.

5. The Local Lemma.

5.1 The Lemma.

5.2 Property B and Multicolored Sets of Real Numbers.

5.3 Lower Bounds for Ramsey Numbers.

5.4 A Geometric Result.

5.5 The Linear Arboricity of Graphs.

5.6 Latin Transversals.

5.7 The Algorithmic Aspect.

5.8 Exercises.

The Probabilistic Lens: Directed Cycles.

6. Correlation Inequalities.

6.1 The Four Functions Theorem of Ahlswede.

and Daykin.

6.2 The FKG Inequality.

6.3 Monotone Properties.

6.4 Linear Extensions of Partially Ordered Sets.

6.5 Exercises.

The Probabilistic Lens: Turán’s Theorem.

7. Martingales and Tight Concentration.

7.1 Definitions.

7.2 Large Deviations.

7.3 Chromatic Number.

7.4 Two General Settings.

7.5 Four Illustrations.

7.6 Talagrand’s Inequality.

7.7 Applications of Talagrand’s Inequality.

7.8 KimVu.

7.9 Exercises.

The Probabilistic Lens: Weierstrass Approximation Theorem.

8. The Poisson Paradigm.

8.1 The Janson Inequalities.

8.2 The Proofs.

8.3 Brun’s Sieve.

8.4 Large Deviations.

8.5 Counting Extensions.

8.6 Counting Representations.

8.7 Further Inequalities.

8.8 Exercises.

The Probabilistic Lens: Local Coloring.

9. Pseudorandomness.

9.1 The Quadratic Residue Tournaments.

9.2 Eigenvalues and Expanders.

9.3 Quasi Random Graphs.

9.4 Exercises.

The Probabilistic Lens: Random Walks.


10 Random Graphs.

10.1 Subgraphs.

10.2 Clique Number.

10.3 Chromatic Number.

10.4 ZeroOne Laws.

10.5 Exercises.

The Probabilistic Lens: Counting Subgraphs.

11. The Erd" osR.

‘enyi Phase Transition.

11.1 An Overview.

11.2 Three Processes.

11.3 The GaltonWatson Branching Process.

11.4 Analysis of the Poisson Branching Process.

11.5 The Graph Branching Model.

11.6 The Graph and Poisson Processes Compared.

11.7 The Parametrization Explained.

11.8 The Subcritical Regions.

11.9 The Supercritical Regimes.

11.10 The Critical Window.

11.11 Analogies to Classical Percolation Theory.

11.12 Exercises.

The Probabilistic Lens: The Rich Get Richer.

12. Circuit Complexity.

12.1 Preliminaries 318.

12.2 Random Restrictions and BoundedDepth Circuits.

12.3 More on BoundedDepth Circuits.

12.4 Monotone Circuits.

12.5 Formulae.

12.6 Exercises.

The Probabilistic Lens: Maximal Antichains.

13. Discrepancy.

13.1 Basics.

13.2 Six Standard Deviations Suffice.

13.3 Linear and Hereditary Discrepancy.

13.4 Lower Bounds.

13.5 The BeckFiala Theorem.

13.6 Exercises.

The Probabilistic Lens: Unbalancing Lights.

14. Geometry.

14.1 The Greatest Angle among Points in Euclidean Spaces.

14.2 Empty Triangles Determined by Points in the Plane.

14.3 Geometrical Realizations of Sign Matrices.

14.4 QNets and VCDimensions of Range Spaces.

14.5 Dual Shatter Functions and Discrepancy.

14.6 Exercises.

The Probabilistic Lens: Efficient Packing.

15. Codes, Games and Entropy.

15.1 Codes.

15.2 Liar Game.

15.3 Tenure Game.

15.4 Balancing Vector Game.

15.5 Nonadaptive Algorithms.

15.6 Half Liar Game.

15.7 Entropy.

15.8 Exercises.

The Probabilistic Lens: An Extremal Graph.

16. Derandomization.

16.1 The Method of Conditional Probabilities.

16.2 dWise Independent Random Variables in Small Sample Spaces.

16.3 Exercises.

The Probabilistic Lens: Crossing Numbers, Incidences, Sums and Products.

17. Graph Property Testing.

17.1 Property Testing.

17.2 Testing colorability.

17.3 Szemer ’edi’s Regularity Lemma.

17.4 Testing trianglefreeness.

17.5 Characterizing the testable graph properties.

17.6 Exercises.

The Probabilistic Lens: Tur?an Numbers and Dependent Random Choice.

Appendix A: Bounding of Large Deviations.

A.1 Chernoff Bounds.

A.2 Lower Bounds.

A.3 Exercises.

The Probabilistic Lens: Trianglefree Graphs Have Large Independence Numbers.

Appendix B: Paul Erd" os.

B.1 Papers.

B.2 Conjectures.

B.3 On Erd" os.

B.4 Uncle Paul.


Subject Index.

Author Index.