## The Probabilistic MethodThe leading reference on probabilistic methods in combinatorics-now expanded and updated When it was first published in 1991, The Probabilistic Method became instantly the standard reference on one of the most powerful and widely used tools in combinatorics. Still without competition nearly a decade later, this new edition brings you up to speed on recent developments, while adding useful exercises and over 30% new material. It continues to emphasize the basic elements of the methodology, discussing in a remarkably clear and informal style both algorithmic and classical methods as well as modern applications. The Probabilistic Method, Second 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. New and revised coverage includes: * Several improved as well as new results * A continuous approach to discrete probabilistic problems * Talagrand's Inequality and other novel concentration results * A discussion of the connection between discrepancy and VC-dimension * Several combinatorial applications of the entropy function and its properties * A new section on the life and work of Paul Erdös-the developer of the probabilistic method |

### Contents

TOPICS | 153 |

Bounding of Large Deviations | 263 |

Trianglefree Graphs Have Large Independence Numbers | 272 |

Paul Erdos | 275 |

References | 283 |

295 | |

299 | |

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A₁ algorithm apply asymptotic B₁ binary Boolean function Carole Chapter chips choice circuit coloring combinatorial complete graph completing the proof components compute conjecture consider constant contains Corollary define denote the number digraph disc(A disjoint distribution eigenvalue event exists expected number finite fixed follows given gives graph G Graph Theory Hamiltonian paths hence hypergraph implies independent set indicator random variable induced subgraph inequality integer intersection least Lemma Let G Let H linearity of expectation log₂ lower bound martingale matrix monochromatic mutually independent n-set number of edges number of vertices p₁ pairs Paul Erdős points polynomial positive probability Pr(A Pr[A Pr[B Pr[T Pr[X PROBABILISTIC LENS probabilistic method probability space problem prove R₁ random graph random variable range space result satisfies set of vertices subgraph Suppose tournament triangle two-coloring upper bound v₁ Var[X VC-dimension vector vertex X₁ y₁

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Page ii - USA JAN KAREL LENSTRA Department of Mathematics and Computer Science, Eindhoven University of Technology, Eindhoven, The Netherlands JOEL H.