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The Strange Logic of Random Graphs : Algorithms and Combinatorics - Joel Spencer

The Strange Logic of Random Graphs

Algorithms and Combinatorics

Hardcover

Published: 20th June 2001
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The study of random graphs was begun by Paul Erdos and Alfred Renyi in the 1960s and now has a comprehensive literature. A compelling element has been the threshold function, a short range in which events rapidly move from almost certainly false to almost certainly true. This book now joins the study of random graphs (and other random discrete objects) with mathematical logic. The possible threshold phenomena are studied for all statements expressible in a given language. Often there is a zero-one law, that every statement holds with probability near zero or near one. The methodologies involve probability, discrete structures and logic, with an emphasis on discrete structures. The book will be of interest to graduate students and researchers in discrete mathematics.

From the reviews of the first edition:





"The author ... is a leading expert in random graph theory, and reputed for his expository style. His recent book is again a well-written and exciting text, which I warmly recommend to researchers and graduate students interested in the subject. ... The book has a clear and vivid style, and the material is essentially self-contained, so it is very well-suited for self-study." (Peter Mester, Acta Scientiarum Mathematicarum, Vol. 69, 2003)





"This beautifully written book deals with the fascinating world of random graphs, using a nice blend of techniques coming from combinatorics, probability and mathematical logic, while keeping the treatment self-contained." (Alessandro Berarducci, Mathematical Reviews, Issue 2003 d)

Beginnings
Two Starting Examplesp. 3
A Blend of Probability, Logic and Combinatoricsp. 3
A Random Unary Predicatep. 8
Some Comments on Referencesp. 11
Preliminariesp. 13
What is the Random Graph G(n,p)?p. 13
The Erdos-Renyi Evolutionp. 14
The Appearance of Small Subgraphsp. 15
What is a First Order Theory?p. 15
Extension Statements and Rooted Graphsp. 17
What is a Zero-One Law?p. 18
Almost Sure Theories and Complete Theoriesp. 20
Countable Modelsp. 20
The Ehrenfeucht Gamep. 23
The Rules of the Gamep. 23
Equivalence Classes and Ehrenfeucht Valuep. 27
Connection to First Order Sentencesp. 31
Inside-Outside Strategiesp. 33
The Bridge to Zero-One Lawsp. 37
Other Structuresp. 39
General First Order Structuresp. 39
The Simple Case of Total Orderp. 40
k-Similar Neighborhoodsp. 42
Random Graphs
Very Sparse Graphsp. 49
The Voidp. 50
On the k-th Dayp. 50
On Day $$p. 51
An Excursion into Rooted Treesp. 51
Two Consequencesp. 55
Past the Double Jumpp. 56
Beyond Connectivityp. 57
Limiting Probabilitiesp. 58
A General Result on Limiting Probabilitiesp. 58
In the Beginningp. 59
On the k-th Dayp. 60
At the Threshold of Connectivityp. 61
p. 62
Poisson Childbearingp. 63
Almost Completing the Almost Sure Theoryp. 65
The Combinatorics of Rooted Graphsp. 69
Sparse, Dense, Rigid, Safep. 69
The / Closurep. 73
The Finite Closure Theoremp. 74
The Janson Inequalityp. 79
Extension Statementsp. 80
Counting Extensionsp. 82
Generic Extensionp. 85
The Main Theoremp. 87
The Look-Ahead Strategyp. 87
The Final Movep. 88
The Core Argument (Middle Moves)p. 88
The First Movep. 89
The Original Argumentp. 90
Countable Modelsp. 93
An Axiomatization for T$$p. 93
The Schemap. 93
Completeness Proofp. 93
The Truth Gamep. 95
Countable Modelsp. 97
Constructionp. 97
Uniqueness of the Modelp. 99
Non Uniqueness of the Modelp. 100
A Continuum of Complete Theoriesp. 102
Near Rational Powers of np. 103
Infinitely Many Ups and Downsp. 103
In the Second Order Worldp. 103
Replacing Second Order by First Orderp. 105
Are First Order Properties Natural?p. 107
Existence of Finite Modelsp. 108
Non Separability and Non Convergencep. 109
Representing All Finite Graphsp. 110
Non Separabilityp. 111
Arithmetizationp. 112
Non Convergencep. 113
The Last Thresholdp. 115
Just Past n-$$: The Theory T-$$p. 115
Just Past n-$$: A Zero-One Lawp. 117
Extras
A Dynamic Viewp. 121
More Zero-One Lawsp. 121
Near Irrational Powersp. 121
Dense Random Graphsp. 122
The Limit Functionp. 122
Definitionp. 122
Look-Ahead Functionsp. 123
Well Ordered Discontinuitiesp. 124
Underapproximation sequencesp. 125
Determination in PHp. 127
Stringsp. 131
Models and Languagep. 131
Ehrenfeucht Reduxp. 132
The Rulesp. 132
The Semigroupp. 133
Long Stringsp. 133
Persistent and Transientp. 134
Persistent Stringsp. 136
Random Stringsp. 137
Circular Stringsp. 138
Sparse Unary Predicatep. 140
Stronger Logicsp. 145
Ordered Graphsp. 145
Arithmetizationp. 145
Dance Marathonp. 147
Slow Oscillationp. 148
Existential Monadicp149
Three Final Examplesp. 153
Random Functionsp. 153
Distance Random G(n,p)p. 155
Without Orderp. 155
With Orderp. 157
Random Liftsp. 160
Bibliographyp. 165
Indexp. 167
Table of Contents provided by Publisher. All Rights Reserved.

ISBN: 9783540416548
ISBN-10: 3540416544
Series: Algorithms and Combinatorics
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 168
Published: 20th June 2001
Publisher: Springer-Verlag Berlin and Heidelberg Gmbh & Co. Kg
Country of Publication: DE
Dimensions (cm): 23.5 x 15.5  x 1.91
Weight (kg): 0.38