The Probabilistic Method.pdf

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THE PROBABILISTIC METHOD

THE PROBABILISTIC METHOD Third edition, September 2007, Tel Aviv and New York Noga Alon, School of Mathematics, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, Israel. Joel H. Spencer, Courant Institute of Mathematical Sciences, New York University, New York, USA A Wiley-Interscience Publication JOHN WILEY & SONS, INC. New York / Chichester / Weinheim / Brisbane / Singapore / Toronto

To Nurit and Mary Ann

Contents Dedication Preface Acknowledgments Part I METHODS 1 The Basic Method The Probabilistic Method 1.1 1.2 Graph Theory 1.3 Combinatorics 1.4 Combinatorial Number Theory 1.5 Disjoint Pairs 1.6 Exercises The Probabilistic Lens: The Erd ˝os–Ko–Rado Theorem 2 Linearity of Expectation 2.1 Basics v xiii xv 1 1 3 7 9 10 11 13 15 15 vii

viii CONTENTS Splitting Graphs Two Quickies Balancing Vectors 2.2 2.3 2.4 2.5 Unbalancing Lights 2.6 Without Coin Flips 2.7 Exercises The Probabilistic Lens: Br´egman’s Theorem 3 Alterations Ramsey Numbers Independent Sets 3.1 3.2 3.3 Combinatorial Geometry 3.4 3.5 3.6 Continuous Time 3.7 Packing Recoloring Exercises The Probabilistic Lens: High Girth and High Chromatic Number 4 The Second Moment Basics 4.1 4.2 Number Theory 4.3 More Basics 4.4 Random Graphs 4.5 Clique Number 4.6 Distinct Sums 4.7 4.8 The R¨odl Nibble Exercises The Probabilistic Lens: Hamiltonian Paths 5 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 5.1 5.2 5.3 5.4 5.5 16 18 19 21 22 23 24 27 27 29 30 31 32 35 39 41 43 43 44 47 49 53 54 56 61 63 67 67 70 71 73 74

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