## 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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adjacent algorithm apply arbitrary argument assume assumption asymptotic blue bound called Chapter chips choice choose chosen circuit claim Clearly coloring combinatorial completing the proof components compute conditional conjecture consider constant construction contains copies Corollary corresponding deﬁne denote describe disjoint distinct distribution edges elements equal event example exists expected extension fact ﬁnd ﬁnite ﬁrst ﬁxed follows function given gives graph hence holds implies independent induction inequality integer intersection least Lemma length less linear mathematics mean method monochromatic move mutually Note obtained otherwise pairs Paul points polynomial positive possible precisely probabilistic probability problem Proof prove random graph random variable randomly range result satisﬁes satisfy selected simple space subgraph subsets Suppose taking Theorem Theory tournament triangle vector vertex vertices wins

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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.