Inevitable randomness in discrete mathematics

267 Pages·2009·1.489 MB·English

by  Jozsef Beck

University
L
ECTURE
Series
Volume 49
Inevitable Randomness in
Discrete Mathematics
József Beck
American Mathematical Society

Inevitable Randomness in
Discrete Mathematics

University
L
ECTURE
Series
Volume 49
Inevitable Randomness in
Discrete Mathematics
József Beck
ERMIACAN MΑΓΕΩΜΕAΤTΡHΗEΤΟMΣ AΜTΗICΕΙΣΙΤΩALSOYCTIE
FOUNDED 1888
American Mathematical Society
Providence, Rhode Island
EDITORIAL COMMITTEE
Jerry L. Bona Nigel D. Higson
Eric M. Friedlander (Chair) J. T. Stafford
2000 Mathematics Subject Classification. Primary 60–02, 05–02, 91A46;
Secondary 05D40, 11K38.
For additional informationand updates on this book, visit
www.ams.org/bookpages/ulect-49
Library of Congress Cataloging-in-Publication Data
Beck,J´ozsef.
Inevitablerandomnessindiscretemathematics/Jo´zsefBeck.
p.cm. —(Universitylectureseries;v.49)
Includesbibliographicalreferences.
ISBN978-0-8218-4756-5(alk.paper)
1.Gametheory. 2.Randommeasures. I.Title.
QA269.B336 2009
519.3—dc22
2009011727
Copying and reprinting. Individual readers of this publication, and nonprofit libraries
actingforthem,arepermittedtomakefairuseofthematerial,suchastocopyachapterforuse
in teaching or research. Permission is granted to quote brief passages from this publication in
reviews,providedthecustomaryacknowledgmentofthesourceisgiven.
Republication,systematiccopying,ormultiplereproductionofanymaterialinthispublication
is permitted only under license from the American Mathematical Society. Requests for such
permissionshouldbeaddressedtotheAcquisitionsDepartment,AmericanMathematicalSociety,
201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by
[email protected].
⃝c 2009bytheAmericanMathematicalSociety. Allrightsreserved.
TheAmericanMathematicalSocietyretainsallrights
exceptthosegrantedtotheUnitedStatesGovernment.
PrintedintheUnitedStatesofAmerica.
⃝∞ Thepaperusedinthisbookisacid-freeandfallswithintheguidelines
establishedtoensurepermanenceanddurability.
VisittheAMShomepageathttp://www.ams.org/
10987654321 141312111009
Contents
Preface ix
Part A. Reading the shadows on the wall and formulating a vague
conjecture 1
Chapter 1. Complex systems 3
1. Order and Disorder 3
2. Ideal gases and the Equiprobability Postulate 4
3. Apparent randomness of primes and the Riemann Hypothesis 6
4. Zoo of zeta-functions 10
Chapter 2. Collecting data: Apparent randomness of digit sequences 13
1. Normal numbers 13
2. Continued fraction 14
3. Equidistribution and continued fraction 16
4. More on continued fraction and diophantine approximation 17
Chapter 3. Collecting data: More randomness in number theory 21
1. The Twin Prime Conjecture and Independence 21
2. Finite fields and the congruence Riemann Hypothesis 23
3. Randomness in the two classical lattice point counting problems 24
4. The 3𝑛+1 Conjecture 27
5. Primes represented by individual quadratic forms 28
6. Continued fraction: The length of the period for quadratic irrationals 34
Chapter 4. Laplace and the Principle of Insufficient Reason 37
1. Introduction 37
2. Randomness and probability 40
3. Complexity and randomness of individual sequences 43
4. Formulating a vague probabilistic conjecture 44
5. Limitations of the SLG Conjecture 47
Chapter 5. Collecting proofs for the SLG Conjecture 49
1. When independence is more or less plausible 49
2. Another Central Limit Theorem: “Randomness of the square root of 2” 53
3. Problems without apparent independence: Inevitable irregularities—
an illustration of the Solid-Liquid-Gas Conjecture 58
v
vi CONTENTS
Part B. More evidence for the SLG Conjecture: Exact solutions in
real game theory 67
Chapter 6. Ramsey Theory and Games 69
1. The usual quick jump from easy to hard 69
2. A typical hard problem: Ramsey Numbers. A case of Inaccessible
Data 71
3. Another hard problem: Ramsey Games 74
4. Weak Ramsey Games: Here we know the right order of magnitude! 76
5. Proof of the lower bound in (6.10) 77
6. An interesting detour: Extremal Hypergraphs of the Erd˝os–Selfridge
theorem and the Move Number 81
7. Concluding note on off-diagonal Ramsey Numbers 86
Chapter 7. Practice session (I): More on Ramsey Games and strategies 89
1. Halving strategy 89
2. Switching to the complete bipartite graph 𝐾 . Completing the proof
𝑛,𝑙
of (6.10) 92
3. Understanding the threshold in (6.10). Random Play Intuition 93
4. Move Number 94
5. An interesting detour: Game vs. Ramsey 96
Chapter 8. Practice session (II): Connectivity games and more strategies 99
1. Lehman’s theorem 99
2. Erd˝os’s random graph intuition 101
3. Forcing isolated points 102
4. The Chva´tal–Erd˝os proof: Quick greedy building 103
5. Slow building via blocking: The Transversal Hypergraph Method 106
6. Proof of Proposition 8.3 108
Chapter 9. What kind of games? 111
1. Introduction 111
2. The Tic-Tac-Toe family 113
3. Where is the breaking point from draw to win? A humiliating gap in
our knowledge! 117
4. First simplification: Replacing ordinary Win with Weak Win 118
Chapter10. Exactsolutionsofgames: UnderstandingviatheEquiprobability
Postulate 123
1. Another simplification: Switching from Maker-Breaker games to
Cut-and-Choose games 123
2. Sim and other Clique Games on graphs 125
3. The concentration of random variables in general 126
4. How does the Equiprobability Postulate enter real game theory? 129
5. Rehabilitation of Laplace? 133
Chapter 11. Equiprobability Postulate with Constraints (Endgame Policy) 135
1. Introduction 135
2. Modifying the Equiprobability Postulate with an Endgame Policy 136
3. Going back to 1-dimensional goals 138
CONTENTS vii
4. Finding the correct form of the Biased Weak Win Conjecture when
Maker is the topdog 139
5. Coalition Games 142
6. Vague Equiprobability Conjecture 143
7. Philosophical speculations on a probabilistic paradigm 145
Chapter 12. Constraints and Threshold Clustering 147
1. What are the constraints of ordinary win? What are the constraints
of Ramsey Theory? 147
2. Delicate win or delicate draw? A wonderful question! 151
3. Threshold Clustering 152
Chapter 13. Threshold Clustering and a few bold conjectures 155
1. Examples 155
2. What to do next? Searching for simpler problems 161
Part C. New evidence: Games and Graphs, the Surplus, and the
Square Root Law 163
Chapter 14. Yet another simplification: Sparse hypergraphs and the Surplus 165
1. Row-Column Games 165
2. Exact solutions 169
3. The Core-Density and the Surplus 171
4. Remarks 173
5. Regular graphs—local randomness 174
6. How sharp is Theorem 1? 175
Chapter 15. Is Surplus the right concept? (I) 177
1. Socialism does work on graphs! 177
2. Do-It-First Lead 179
3. Monopoly 179
4. Shutout 180
5. Inevitable Shutout 183
Chapter 16. Is Surplus the right concept? (II) 185
1. The Move Number 185
2. Discrepancy and variance 188
3. Summary 189
Chapter 17. Working with a game-theoretic Partition Function 193
1. Introduction 193
2. The lower bound 195
3. Some underdog versions of Proposition 17.3 197
Chapter 18. An attempt to save the Variance 203
1. Introduction 203
2. An alternative approach 204
Chapter19. ProofofTheorem1: Combiningthevariancewithanexponential
sum 209
1. Defining a complicated potential function 209
viii CONTENTS
2. Global balancing 212
3. Average argument 215
Chapter 20. Proof of Theorem 2: The upper bound 219
1. Can we use the Local Lemma in games? 219
2. Danger function: Big-Game & small-game decomposition 220
Chapter 21. Conclusion (I): More on Theorem 1 227
1. Threshold Clustering: Generalizations of Theorem 1 227
2. When threshold clustering fails: Shutout games 230
3. Last remark about Theorem 1 233
Chapter 22. Conclusion (II): Beyond the SLG Conjecture 237
1. Wild speculations: Is it true that most unrestricted do-it-first games
are unknowable? 237
2. Weak Win and Infinite Draw 240
Dictionary of the Phrases and Concepts 245
References 247