The Probabilistic Method (Hardcover, 3)

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. ...