The Strange Logic of Random Graphs

Joel Spencer (auth.)3540416544, 9783540416548

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.

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Table of contents :
Front Matter….Pages I-X
Front Matter….Pages 1-1
Two Starting Examples….Pages 3-12
Preliminaries….Pages 13-21
The Ehrenfeucht Game….Pages 23-45
Front Matter….Pages 47-47
Very Sparse Graphs….Pages 49-67
The Combinatorics of Rooted Graphs….Pages 69-77
The Janson Inequality….Pages 79-86
The Main Theorem….Pages 87-91
Countable Models….Pages 93-102
Near Rational Powers of n ….Pages 103-118
Front Matter….Pages 119-119
A Dynamic View….Pages 121-129
Strings….Pages 131-144
Stronger Logics….Pages 145-151
Three Final Examples….Pages 153-163
Back Matter….Pages 165-168