Probabilistic Method

ISBN-10: 0470170204
ISBN-13: 9780470170205
Edition: 3rd 2008
List price: $148.00 Buy it from $41.12 Rent it from $36.33
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Description: This book is the leading reference on probabilistic methods in combinatorics, and there is no direct competition. New to the Third Edition is a chapter devoted to Graph Property Testing and the included sections are Graph Property Testing; Testing  More...

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Book details

List price: $148.00
Edition: 3rd
Copyright year: 2008
Publisher: John Wiley & Sons, Incorporated
Publication date: 8/11/2008
Binding: Hardcover
Pages: 376
Size: 6.25" wide x 9.50" long x 1.00" tall
Weight: 1.672
Language: English

This book is the leading reference on probabilistic methods in combinatorics, and there is no direct competition. New to the Third Edition is a chapter devoted to Graph Property Testing and the included sections are Graph Property Testing; Testing Colorability; Szemeredi's Regularity Lemma; Testing Triangle-Freeness; and Characterizing the Testable Graph Properties. New sections have also been added on Percolation, Webgraphs, and Chernoff Bounds. A substantial revision has been made to the Double Jump section. The Probabilistic Method, Third Edition begins with basic techniques that use expectation and variance, as well as the more recent martingales and correlation inequalities, then explores areas where probabilistic techniques proved successful, including discrepancy and random graphs as well as cutting-edge topics in theoretical computer science. A series of proofs, or "probabilistic lenses," are interspersed throughout the book, offering added insight into the application of the probabilistic approach. The number of exercises included in the third edition has been almost doubled from that of the second edition, and hints and/or answers to some of the exercises are provided.

Preface
Acknowledgments
Methods
The Basic Method
The Probabilistic Method
Graph Theory
Combinatorics
Combinatorial Number Theory
Disjoint Pairs
Exercises
The Probabilistic Lens: The Erdos-Ko-Rado Theorem
Linearity of Expectation
Basics
Splitting Graphs
Two Quickies
Balancing Vectors
Unbalancing Lights
Without Coin Flips
Exercises
The Probabilistic Lens: Bregman's Theorem
Alterations
Ramsey Numbers
Independent Sets
Combinatorial Geometry
Packing
Recoloring
Continuous Time
Exercises
The Probabilistic Lens: High Girth and High Chromatic Number
The Second Moment
Basics
Number Theory
More Basics
Random Graphs
Clique Number
Distinct Sums
The Rodl Nibble
Exercises
The Probabilistic Lens: Hamiltonian Paths
The Local Lemma
The Lemma
Property B and Multicolored Sets of Real Numbers
Lower Bounds for Ramsey Numbers
A Geometric Result
The Linear Arboricity of Graphs
Latin Transversals
The Algorithmic Aspect
Exercises
The Probabilistic Lens: Directed Cycles
Correlation Inequalities
The Four Functions Theorem of Ahlswede and Daykin
The FKG Inequality
Monotone Properties
Linear Extensions of Partially Ordered Sets
Exercises
The Probabilistic Lens: Turan's Theorem
Martingales and Tight Concentration
Definitions
Large Deviations
Chromatic Number
Two General Settings
Four Illustrations
Talagrand's Inequality
Applications of Talagrand's Inequality
Kim-Vu Polynomial Concentration
Exercises
The Probabilistic Lens: Weierstrass Approximation Theorem
The Poisson Paradigm
The Janson Inequalities
The Proofs
Brun's Sieve
Large Deviations
Counting Extensions
Counting Representations
Further Inequalities
Exercises
The Probabilistic Lens: Local Coloring
Pseudorandomness
The Quadratic Residue Tournaments
Eigenvalues and Expanders
Quasirandom Graphs
Exercises
The Probabilistic Lens: Random Walks
Topics
Random Graphs
Subgraphs
Clique Number
Chromatic Number
Zero-One Laws
Exercises
The Probabilistic Lens: Counting Subgraphs
The Erdos-Renyi Phase Transition
An Overview
Three Processes
The Galton-Watson Branching Process
Analysis of the Poisson Branching Process
The Graph Branching Model
The Graph and Poisson Processes Compared
The Parametrization Explained
The Subcritical Regimes
The Supercritical Regimes
The Critical Window
Analogies to Classical Percolation Theory
Exercises
The Probabilistic Lens: The Rich Get Richer
Circuit Complexity
Preliminaries
Random Restrictions and Bounded-Depth Circuits
More on Bounded-Depth Circuits
Monotone Circuits
Formulae
Exercises
The Probabilistic Lens: Maximal Antichains
Discrepancy
Basics
Six Standard Deviations Suffice
Linear and Hereditary Discrepancy
Lower Bounds
The Beck-Fiala Theorem
Exercises
The Probabilistic Lens: Unbalancing Lights
Geometry
The Greatest Angle Among Points in Euclidean Spaces
Empty Triangles Determined by Points in the Plane
Geometrical Realizations of Sign Matrices
[epsilon]-Nets and VC-Dimensions of Range Spaces
Dual Shatter Functions and Discrepancy
Exercises
The Probabilistic Lens: Efficient Packing
Codes, Games and Entropy
Codes
Liar Game
Tenure Game
Balancing Vector Game
Nonadaptive Algorithms
Half Liar Game
Entropy
Exercises
The Probabilistic Lens: An Extremal Graph
Derandomization
The Method of Conditional Probabilities
d-Wise Independent Random Variables in Small Sample Spaces
Exercises
The Probabilistic Lens: Crossing Numbers, Incidences, Sums and Products
Graph Property Testing
Property Testing
Testing Colorability
Szemeredi's Regularity Lemma
Testing Triangle-Freeness
Characterizing the Testable Graph Properties
Exercises
The Probabilistic Lens: Turan Numbers and Dependent Random Choice
Bounding of Large Deviations
Chernoff Bounds
Lower Bounds
Exercises
The Probabilistic Lens: Triangle-Free Graphs Have Large Independence Numbers
Paul Erdos
Papers
Conjectures
On Erdos
Uncle Paul
References
Author Index
Subject Index

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