## The Probabilistic MethodThe leading reference on probabilistic methods in combinatorics-nowexpanded and updated When it was first published in 1991, The Probabilistic Methodbecame instantly the standard reference on one of the most powerfuland widely used tools in combinatorics. Still without competitionnearly a decade later, this new edition brings you up to speed onrecent developments, while adding useful exercises and over 30% newmaterial. It continues to emphasize the basic elements of themethodology, discussing in a remarkably clear and informal styleboth algorithmic and classical methods as well as modernapplications. The Probabilistic Method, Second Edition begins with basictechniques that use expectation and variance, as well as the morerecent martingales and correlation inequalities, then exploresareas where probabilistic techniques proved successful, includingdiscrepancy and random graphs as well as cutting-edge topics intheoretical computer science. A series of proofs, or "probabilisticlenses," are interspersed throughout the book, offering addedinsight into the application of the probabilistic approach. New andrevised coverage includes: * Several improved as well as new results * A continuous approach to discrete probabilistic problems * Talagrand's Inequality and other novel concentrationresults * A discussion of the connection between discrepancy andVC-dimension * Several combinatorial applications of the entropy function andits properties * A new section on the life and work of Paul Erdös-thedeveloper 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.