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.

Random Graphs are a research field of major importance in discrete mathematics. This is an excellent book by one of the top researchers in this field Includes supplementary material: sn.pub/extras

Autorentext

Erich Graedel is a Professor of Mathematical Foundations of Computer Science at the University of Technology Aachen. His research interests include algorithms, complexity, and logic in computer science. Phokion G. Kolaitis is a professor of computer science at the University of California, Santa Cruz. His current research interests include logic in computer science, computational complexity, and database theory. He earned a Diploma in Mathematics from the University of Athens, Greece in 1973, and a Ph.D. in Mathematics from the University of California, Los Angeles in 1978. Before joining UC Santa Cruz in 1988, he served as an L.E. Dickson Instructor of Mathematics at the University of Chicago, a faculty member at Occidental College, a visiting faculty member at Stanford University, and a visiting scientist at the IBM Almaden Research Center. Kolaitis was awarded a Guggenheim Fellowship during 1993-94. In 1995, he received an Excellence in Teaching Award by the graduating computer science and computer engineering students at UC Santa Cruz. Leonid Libkin received his PhD from the University of Pennsylvania and is currently Professor of Computer Science at the University of Toronto. His main research interests include databases and applications of logic in computer science. Maarten Marx is an associate professor at the Vrije Universiteit Amsterdam. His research interests are in modal and algebraic logic. Joel Spencer is a Professor of Mathematics and Computer Scienceat the Courant Institute, New York University. His research interests lie in interface between Discrete Mathematics and Theoretical Computer Science, most particularly with the Probabilistic Method as developed by Paul Erdos. Moshe Y. Vardi is a Noah Harding Professor of Computer Science and Chair of Computer Science at Rice University. Prior to joining Rice in 1993, he was at the IBM Almaden Research Center, where he managed the Mathematics and Related Computer Science Department. His research interests include database systems, computational-complexity theory, multi-agent systems, and design specification and verification. Vardi received his Ph.D. from the Hebrew University of Jerusalem in 1981. He is the author and co-author of over 120 technical papers, as well as a book titled "Reasoning about Knowledge". Vardi is the recipient of 3 IBM Outstanding Innovation Awards. He is an editor of several international journals and is a Fellow of the Association of Computing Machinery. Yde Venema studied mathematics; in 1992, he received a PhD in Logic with the dissertation `Many-Dimensional Modal Logic'. He is currently a Research Fellow of the Royal Netherlands Academy of Arts and Sciences and an assistant professor at the Institute for Logic, Language and Computation of the University of Amsterdam. His research interests include modal and temporal logic, algebraic logic, and applications of logic in linguistics and computer science. Scott Weinstein is Professor of Computer Science, Mathematics, and Philosophy at the University of Pennsylvania. His research interests include logic in computer science and the philosphy of mathematics.

Inhalt
I. Beginnings.- 0. Two Starting Examples.- 1. Preliminaries.- 2. The Ehrenfeucht Game.- II. Random Graphs.- 3. Very Sparse Graphs.- 4. The Combinatorics of Rooted Graphs.- 5. The Janson Inequality.- 6. The Main Theorem.- 7. Countable Models.- 8. Near Rational Powers of n.- III. Extras.- 9. A Dynamic View.- 10. Strings.- 11. Stronger Logics.- 12. Three Final Examples.