## 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 |

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### 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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algorithm apply asymptotic binary Boolean function Carole Chapter chips choice circuit coloring combinatorial complete graph completing the proof components compute conditional probability conjecture consider constant contains Corollary deﬁne deﬁnition denote the number digraph disc(A disjoint distribution efﬁcient eigenvalue Erdos Erdt'is event exists expected number ﬁnal ﬁnd ﬁnite ﬁrst ﬁxed ﬂips follows given gives graph G Graph Theory Hadamard matrix Hamiltonian paths hence hypergraph implies independent set indicator random variable induced subgraph inequality inﬁnite integer intersection least Lemma Let G I Let H linearity of expectation lower bound martingale matrix monochromatic mutually independent n-set number of edges pairs Paul Paul Erdos points polynomial positive probability Pr(A Pr[B Pr[X Probabilistic Lens probabilistic method probability space problem prove random graph range space result satisﬁes set of vertices speciﬁc subgraph sufﬁces sufﬁciently large Suppose tournament triangle two-coloring upper bound VC-dimension vector vertex

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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.