Wiley Series in Discrete Mathematics and Optimization Fourth Edition THE PROBABILISTIC METHOD NOGA ALON JOEL H.SPENCER WILEY
Contents PREFACE xi ACKNOWLEDGMENTS PARTI METHODS 1 The Basic Method 1.1 The Probabilistic Method,3 Graph Theory.5 Combinatorics,9 1.4 Combinatorial Number Theory,11 1.5 Disjoint Pairs,12 1.6 Independent Sets and List Coloring.13 1.7 Exercises,16 The Erdos-Ko-Rado Theorem,18 2 Linearity of Expectation 19 o
Contents PREFACE xiii ACKNOWLEDGMENTS xv PART I METHODS 1 1 The Basic Method 3 1.1 The Probabilistic Method, 3 1.2 Graph Theory, 5 1.3 Combinatorics, 9 1.4 Combinatorial Number Theory, 11 1.5 Disjoint Pairs, 12 1.6 Independent Sets and List Coloring, 13 1.7 Exercises, 16 The Erd ˝os–Ko–Rado Theorem, 18 2 Linearity of Expectation 19 2.1 Basics, 19 2.2 Splitting Graphs, 20 2.3 Two Quickies, 22 2.4 Balancing Vectors, 23
i进 CONTENTS 2.5 Unbalancing Lights.25 2.6 Without Coin Flips,26 2.7 Exercises.27 Brigman's Theorem,29 3 Alterations 31 3.1 Ramsey Numbers,31 ndent Sets,33 33 ial Geometry,34 5 Packing. Greedy Coloring.36 6 Continuous Time,38 3.7 Exercises.41 High Girth and High Chromatic Number,43 4 The Second Moment 45 4.1 Basics,45 40 Number Theory,46 43 More Basics .49 44 Random Grap 4.5 que Numb Dis 57 The Rodl nib ble,58 4.8 Exercises,64 Hamiltonian Paths,65 5 The Local Lemma 5.1 The lemma 69 5.2 Property B and Multicolored Sets of Real Numbers,72 5.3 Bou nds for Ran 54 The Linear Arb 5.6 of Graphs.76 atin Iransversals,8 5.7 Moser's Fix-It Algorithm,81 5.8 Exercises.87 Directed Cycles,88 6 Correlation Inequalities 89 6.1 The Four Functions Theorem of Ahlswede and Daykin,90 6.2 The FKG Inequality,93 6.3 Monotone Properties,94
viii CONTENTS 2.5 Unbalancing Lights, 25 2.6 Without Coin Flips, 26 2.7 Exercises, 27 Brégman’s Theorem, 29 3 Alterations 31 3.1 Ramsey Numbers, 31 3.2 Independent Sets, 33 3.3 Combinatorial Geometry, 34 3.4 Packing, 35 3.5 Greedy Coloring, 36 3.6 Continuous Time, 38 3.7 Exercises, 41 High Girth and High Chromatic Number, 43 4 The Second Moment 45 4.1 Basics, 45 4.2 Number Theory, 46 4.3 More Basics, 49 4.4 Random Graphs, 51 4.5 Clique Number, 55 4.6 Distinct Sums, 57 4.7 The Rödl nibble, 58 4.8 Exercises, 64 Hamiltonian Paths, 65 5 The Local Lemma 69 5.1 The Lemma, 69 5.2 Property B and Multicolored Sets of Real Numbers, 72 5.3 Lower Bounds for Ramsey Numbers, 73 5.4 A Geometric Result, 75 5.5 The Linear Arboricity of Graphs, 76 5.6 Latin Transversals, 80 5.7 Moser’s Fix-It Algorithm, 81 5.8 Exercises, 87 Directed Cycles, 88 6 Correlation Inequalities 89 6.1 The Four Functions Theorem of Ahlswede and Daykin, 90 6.2 The FKG Inequality, 93 6.3 Monotone Properties, 94
CONTENTS 6.4 Linear Extensions of Partially Ordered Sets,97 6.5 Exercises,99 Turan's Theorem,100 7 Martingales and Tight Concentration 103 7.1 Definitions,103 7.2 Large Deviations,105 7.3 Chromatic Number,107 7.4 Two General Settings,109 75 Four Illustrations,113 7.6 Talagrand's Inequality,116 7.7 Applications of Talas Kim-VuPo Exercises,123 Weierstrass Approximation Theorem,124 8 The Poisson Paradigm 127 8.1 The Janson Inequalities,127 8.2 The Proofs,129 8.3 Brun's Sieve 132 84 Large Deviations,135 只5 Counting Extensions,137 86 Counting 87 Further 142 8.8 Exercises,143 Local Coloring,144 147 9.1 The Quadratic Residue Tournaments,148 9.2 Eigenvalues and Expanders,151 9.3 Quasirandom Graphs,157 94 Szemeredi's Regularity Lemma.165 9.5 Graphons,170 9.6 Exercises,172 Random Walks,174 PARTⅡTOPICS 177 10 Random Graphs 179 10.1 Subgraphs,180
CONTENTS ix 6.4 Linear Extensions of Partially Ordered Sets, 97 6.5 Exercises, 99 Turán’s Theorem, 100 7 Martingales and Tight Concentration 103 7.1 Definitions, 103 7.2 Large Deviations, 105 7.3 Chromatic Number, 107 7.4 Two General Settings, 109 7.5 Four Illustrations, 113 7.6 Talagrand’s Inequality, 116 7.7 Applications of Talagrand’s Inequality, 119 7.8 Kim–Vu Polynomial Concentration, 121 7.9 Exercises, 123 Weierstrass Approximation Theorem, 124 8 The Poisson Paradigm 127 8.1 The Janson Inequalities, 127 8.2 The Proofs, 129 8.3 Brun’s Sieve, 132 8.4 Large Deviations, 135 8.5 Counting Extensions, 137 8.6 Counting Representations, 139 8.7 Further Inequalities, 142 8.8 Exercises, 143 Local Coloring, 144 9 Quasirandomness 147 9.1 The Quadratic Residue Tournaments, 148 9.2 Eigenvalues and Expanders, 151 9.3 Quasirandom Graphs, 157 9.4 Szemerédi’s Regularity Lemma, 165 9.5 Graphons, 170 9.6 Exercises, 172 Random Walks, 174 PART II TOPICS 177 10 Random Graphs 179 10.1 Subgraphs, 180
CONTENTS 10.2 Clique Number,183 10.3 Chromatic Number.184 10.4 Zero-One Laws,186 105 Exercises 193 Counting Subgraphs,195 11 The Erdos-Renyi Phase Transition 197 11.1 An Overview.197 11.2 c 3 9 -Watson Branching Process,201 11. Analysis of the Poisson Branching Process.202 11.5 The Graph Branching Model,204 11.6 The Graph and Poisson Processes Compared,205 11.7 The Parametrization Explained.207 118 The Subcritical Regions.208 11.9 The Supercritical Regimes,209 11.10 The Cri ical winde 212 11.12 Exercises,219 Long paths in the supercritical regime,220 12 Circuit Complexity 223 12.1 Preliminaries,223 12.2 Random Restrictions and Bounded-Depth Circuits,225 12.3 More on Bounded-Depth Circuits,229 12.4 Monotone Circuits.232 12.5 Formulae.235 12.6 Exercises,236 Maximal Antichains,237 13 Discrepancy 239 13.1 Basics,239 13. 2 Six Standard Deviations Suffice,241 13. 3 Linear and Hereditary Discrepancy,245 13.4 Lower Bounds.248 13.5 The Beck-Fiala Theorem,250 13.6 Exercises,251 Unbalaneing Lights,253 14 Geometry 255 14.1 The Greatest Angle Among Points in Euclidean Spaces,256
x CONTENTS 10.2 Clique Number, 183 10.3 Chromatic Number, 184 10.4 Zero–One Laws, 186 10.5 Exercises, 193 Counting Subgraphs, 195 11 The Erdos–Rényi Phase Transition 197 ˝ 11.1 An Overview, 197 11.2 Three Processes, 199 11.3 The Galton–Watson Branching Process, 201 11.4 Analysis of the Poisson Branching Process, 202 11.5 The Graph Branching Model, 204 11.6 The Graph and Poisson Processes Compared, 205 11.7 The Parametrization Explained, 207 11.8 The Subcritical Regions, 208 11.9 The Supercritical Regimes, 209 11.10 The Critical Window, 212 11.11 Analogies to Classical Percolation Theory, 214 11.12 Exercises, 219 Long paths in the supercritical regime, 220 12 Circuit Complexity 223 12.1 Preliminaries, 223 12.2 Random Restrictions and Bounded-Depth Circuits, 225 12.3 More on Bounded-Depth Circuits, 229 12.4 Monotone Circuits, 232 12.5 Formulae, 235 12.6 Exercises, 236 Maximal Antichains, 237 13 Discrepancy 239 13.1 Basics, 239 13.2 Six Standard Deviations Suffice, 241 13.3 Linear and Hereditary Discrepancy, 245 13.4 Lower Bounds, 248 13.5 The Beck–Fiala Theorem, 250 13.6 Exercises, 251 Unbalancing Lights, 253 14 Geometry 255 14.1 The Greatest Angle Among Points in Euclidean Spaces, 256
CONTENTS 14.2 Empty Triangles Determined by Points in the Plane,257 143 n Matrices,259 14.4 61 14.5 Dual Shatter Functions and Discrepancy.260 14.6 Exercises.269 Efficient Packing,270 15 Codes,Games,and Entropy 273 151 Codes 273 15.2 Liar Game,276 53 lenure Game,278 15.4 Balancing Vector Game,279 15.5 Nonadaptive Algorithms,281 15.6 Half Liar Game,282 15.7 Entropy,284 15.8 Exercises,289 An Extremal Graph,291 16 Derandomization 293 16.1 The Method of Conditional Probabilities,293 16.2 d-Wise Independent Random Variables in Small Sample Spaces,297 16.3 Exercises,302 Crossing Numbers,Incidences,Sums and Products,303 17 Graph Property Testing 307 17.1 Property Testing.307 17.2 Testing Colorability,308 17.3 Testing Triangle-Freeness,312 17.4 Characterizing the Testable Graph Properties.314 17.5 Exercises,316 Turdn Numbers and Dependent Random Choice,317 Appendix A Bounding of Large Deviations 321 A.1 Chernoff Bounds,321 A.2 Lower Bounds, 330 A.3 Exercises.334 Triangle-Free Graphs Have Large Independence Numbers,336
CONTENTS xi 14.2 Empty Triangles Determined by Points in the Plane, 257 14.3 Geometrical Realizations of Sign Matrices, 259 14.4 𝜖-Nets and VC-Dimensions of Range Spaces, 261 14.5 Dual Shatter Functions and Discrepancy, 266 14.6 Exercises, 269 Efficient Packing, 270 15 Codes, Games, and Entropy 273 15.1 Codes, 273 15.2 Liar Game, 276 15.3 Tenure Game, 278 15.4 Balancing Vector Game, 279 15.5 Nonadaptive Algorithms, 281 15.6 Half Liar Game, 282 15.7 Entropy, 284 15.8 Exercises, 289 An Extremal Graph, 291 16 Derandomization 293 16.1 The Method of Conditional Probabilities, 293 16.2 d-Wise Independent Random Variables in Small Sample Spaces, 297 16.3 Exercises, 302 Crossing Numbers, Incidences, Sums and Products, 303 17 Graph Property Testing 307 17.1 Property Testing, 307 17.2 Testing Colorability, 308 17.3 Testing Triangle-Freeness, 312 17.4 Characterizing the Testable Graph Properties, 314 17.5 Exercises, 316 Turán Numbers and Dependent Random Choice, 317 Appendix A Bounding of Large Deviations 321 A.1 Chernoff Bounds, 321 A.2 Lower Bounds, 330 A.3 Exercises, 334 Triangle-Free Graphs Have Large Independence Numbers, 336
CONTENTS Appendix B Paul Erdos 339 B.1 Papers,339 B.2 Conjectures,341 B.3 On Erd6s.342 B.4 Uncle Paul,343 The Rich Get Richer,346 Appendix C Hints to Selected Exercises 349 REFERENCES 355 AUTHOR INDEX 367 SUBJECT INDEX 371
xii CONTENTS Appendix B Paul Erdos 339 ˝ B.1 Papers, 339 B.2 Conjectures, 341 B.3 On Erdos, 342 ˝ B.4 Uncle Paul, 343 The Rich Get Richer, 346 Appendix C Hints to Selected Exercises 349 REFERENCES 355 AUTHOR INDEX 367 SUBJECT INDEX 371
Preface The probabilistic method is one of the most powerful and widely used tools applied in entis the importan The s in theo between mathematicsan ence suggests an and this is the approach we tried to adopt in this book.The book thus includes a discussion of algorithmic techniques together with a study of the classical method as well as the modern tools applied in it.The first part of the book contains a description of the tools applied in probabilistic arguments.including the basic techniques that use expectation and variance,as well as the more recent applications of martingales and e elation inequalities.The second part includes a s udy of various to s in es have cces sful This random grap s,as we on several ar oretical ompute Circuit Complexity,Cor Geometry,Graph Property Testing and Derandomization of randomized algorithms.Scattered between the chapters are gems described under the heading"The Probabilistic Lens."These are elegant proofs that are not necessarily related to the chapters after which they appear and can usually be read separately. The basic probabilistic method can be described as follows:In order to prove the ce of a comhinalorial structure with certain propertes,we construc w tha ta randomly ch t in this s desired prope es with whocontribu sitive pro ability.This ethod was initiated by Paul Erdos ted so mu o its development over a 50-year period that it seems appro priate to call it"The Erdos Method."His contribution can be measured not only by his numerous deep results in the subject but also by the many intriguing problems and conjectures posed by him that stimulated a big portion of the research in the area. It seems impossible to write an encyclopedic book on the probabilistic method:too many recent interesting results apply probabilistic arguments,and we do not even try
Preface The probabilistic method is one of the most powerful and widely used tools applied in combinatorics. One of the major reasons for its rapid development is the important role of randomness in theoretical computer science and in statistical physics. The interplay between discrete mathematics and computer science suggests an algorithmic point of view in the study of the probabilistic method in combinatorics, and this is the approach we tried to adopt in this book. The book thus includes a discussion of algorithmic techniques together with a study of the classical method as well as the modern tools applied in it. The first part of the book contains a description of the tools applied in probabilistic arguments, including the basic techniques that use expectation and variance, as well as the more recent applications of martingales and correlation inequalities. The second part includes a study of various topics in which probabilistic techniques have been successful. This part contains chapters on discrepancy and random graphs, as well as on several areas in theoretical computer science: Circuit Complexity, Computational Geometry, Graph Property Testing and, Derandomization of randomized algorithms. Scattered between the chapters are gems described under the heading “The Probabilistic Lens.” These are elegant proofs that are not necessarily related to the chapters after which they appear and can usually be read separately. The basic probabilistic method can be described as follows: In order to prove the existence of a combinatorial structure with certain properties, we construct an appropriate probability space and show that a randomly chosen element in this space has the desired properties with positive probability. This method was initiated by Paul Erdos, ˝ who contributed so much to its development over a 50-year period that it seems appropriate to call it “The Erdos Method.” His contribution can be measured not only by ˝ his numerous deep results in the subject but also by the many intriguing problems and conjectures posed by him that stimulated a big portion of the research in the area. It seems impossible to write an encyclopedic book on the probabilistic method; too many recent interesting results apply probabilistic arguments, and we do not even try
xiv PREFACE to mention all of them.Our emphasis is on methodology.and we thus try to describe the ideas,and not always to give the best possible results if these are too technical to allow a clear presentation.Many of the results are asymptotic,and we use the standard asymptotic notation:for two functions f and g,we write f=O(g)iff scg for all sufficiently large values of the variables of the two functions,where c is an absolute positive constant.We write f=(g)if g=o(f)and f=e(g)if f =O(g) andf=().If the limit of the ratiof/g tends to roas the variables of the function tend to in y we writef=o(g).Fina g denotes thatf=(1+(1)g:that f/g tends to I when the variables tend to infinity.Each chapter ends with a list of exercises.The more difficult ones are marked by (*)The exercises enable readers to check their understanding of the material and also provide the possibility of using the book as a textbook. This is the fourth edition of the book;it contains several improved results and covers various additional topics that developed extensively during the last few years. A breakthrough approach to the Local Lemma is described in Chapter 5.A new algorithmic sented app ach to the"six standard deviations"result in disc oter 12.A novel proof for the study of Property B.ba ed o oring.appears in Chapter 3.In all the above ses,the algorith mic proofs provide essentially new arguments for the existence of the desired objects A new,short section on graph limits has been added to Chapter 9.A technique for counting independent sets in graphs and its application in a graph coloring problem is described in Chapter 1.Further additions include a new Probabilistic Lens,several additional exercises,and a new appendix with hints to selected exercises. As in the previous editions,it is a special pleasure to thank our wives.Nurit and Mary Ann.Their patience,understanding,and encouragement have been key ingre dients in the success of this enterprise. oga Alor JoelH.Spencer Tel Aviv and New York,2015
xiv PREFACE to mention all of them. Our emphasis is on methodology, and we thus try to describe the ideas, and not always to give the best possible results if these are too technical to allow a clear presentation. Many of the results are asymptotic, and we use the standard asymptotic notation: for two functions f and g, we write f = O(g) if f ≤ cg for all sufficiently large values of the variables of the two functions, where c is an absolute positive constant. We write f = Ω(g) if g = O(f) and f = Θ(g) if f = O(g) and f = Ω(g). If the limit of the ratio f ∕g tends to zero as the variables of the functions tend to infinity we write f = o(g). Finally, f ∼ g denotes that f = (1 + o(1))g; that is, f ∕g tends to 1 when the variables tend to infinity. Each chapter ends with a list of exercises. The more difficult ones are marked by (∗). The exercises enable readers to check their understanding of the material and also provide the possibility of using the book as a textbook. This is the fourth edition of the book; it contains several improved results and covers various additional topics that developed extensively during the last few years. A breakthrough approach to the Local Lemma is described in Chapter 5. A new algorithmic approach to the “six standard deviations” result in discrepancy theory is presented in Chapter 12. A novel proof for the study of Property B, based on a random greedy coloring, appears in Chapter 3. In all the above cases, the algorithmic proofs provide essentially new arguments for the existence of the desired objects. A new, short section on graph limits has been added to Chapter 9. A technique for counting independent sets in graphs and its application in a graph coloring problem is described in Chapter 1. Further additions include a new Probabilistic Lens, several additional exercises, and a new appendix with hints to selected exercises. As in the previous editions, it is a special pleasure to thank our wives, Nurit and Mary Ann. Their patience, understanding, and encouragement have been key ingredients in the success of this enterprise. Noga Alon Joel H. Spencer Tel Aviv and New York, 2015
Acknowledgments We are very grateful to all our students and colleagues who contributed to the creation m Thes dgu eich.O er,F rzysztof Choromans Felhem.Naomi Feldheim,Asaf ber Florescu,Lior Gishb Harel,Danny Hefetz,Timo Hirscher,Rani Hod,Mihyun Kang,Joel Lewis,Nati Linial,Guy Moshkovitz,Dhruv Mubayi,Tahl Nowik,Roberto Oliveira,Ron Peled, Will Perkins,Oliver Riordan,Guy Rutenberg.Jeffrey Shallit,Asaf Shapira,Clara Shikhelman,Philipp Sprussel,Aravind Srinivasan,John Steinberger,Elmar Teufl, Shai Vardi,Amit Weinstein,Jed Yang.Mariano Zelke and Yufei Zhao,who pointed out various inaccuracies and misprints,and su uggested in ovements in the senta tion as well as in the Ne edles sibility for the rem s3uoM3uKgu1ouT3do叫平oj qsuodsa0RMse'saY solely ours
Acknowledgments We are very grateful to all our students and colleagues who contributed to the creation of this fourth edition through joint research, helpful discussions, and useful comments. These include Simon Blackburn, Miklós Bóna, Steve Cook, Ehud Friedgut, Oded Goldreich, Omri Ben-Eliezer, Krzysztof Choromanski, Oliver Cooley, Ohad Feldheim, Naomi Feldheim, Asaf Ferber, Laura Florescu, Lior Gishbboliner, Matan Harel, Danny Hefetz, Timo Hirscher, Rani Hod, Mihyun Kang, Joel Lewis, Nati Linial, Guy Moshkovitz, Dhruv Mubayi, Tahl Nowik, Roberto Oliveira, Ron Peled, Will Perkins, Oliver Riordan, Guy Rutenberg, Jeffrey Shallit, Asaf Shapira, Clara Shikhelman, Philipp Sprüssel, Aravind Srinivasan, John Steinberger, Elmar Teufl, Shai Vardi, Amit Weinstein, Jed Yang, Mariano Zelke and Yufei Zhao, who pointed out various inaccuracies and misprints, and suggested improvements in the presentation as well as in the results. Needless to say, the responsibility for the remaining mistakes, as well as the responsibility for the (hopefully not many) new ones, is solely ours