The Probabilistic Method

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 Erdos Prize (1989), the Polya Prize (2000), the Landau Prize (2005), and the Godel 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.

Dedication. Preface. Acknowledgments. PART I. METHODS. 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: Bregman'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 Rodl 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: Turan'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. PART II. TOPICS. 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. References. Subject Index. Author Index.