This volume contains the accounts of papers delivered at the Nato Advanced Study Institute on Finite and Infinite Combinatorics in Sets and Logic held at the Banff Centre, Alberta, Canada from April 21 to May 4, 1991. As the title suggests the meeting brought together workers interested in the interplay between finite and infinite combinatorics, set theory, graph theory and logic. It used to be that infinite set theory, finite combinatorics and logic could be viewed as quite separate and independent subjects. But more and more those disciplines grow together and become interdependent of each other with ever more problems and results appearing which concern all of those disciplines. I appreciate the financial support which was provided by the N. A. T. O. Advanced Study Institute programme, the Natural Sciences and Engineering Research Council of Canada and the Department of Mathematics and Statistics of the University of Calgary. 11l'te meeting on Finite and Infinite Combinatorics in Sets and Logic followed two other meetings on discrete mathematics held in Banff, the Symposium on Ordered Sets in 1981 and the Symposium on Graphs and Order in 1984. The growing inter-relation between the different areas in discrete mathematics is maybe best illustrated by the fact that many of the participants who were present at the previous meetings also attended this meeting on Finite and Infinite Combinatorics in Sets and Logic.

This volume contains the accounts of papers delivered at the Nato Advanced Study Institute on Finite and Infinite Combinatorics in Sets and Logic held at the Banff Centre, Alberta, Canada from April 21 to May 4, 1991. As the title suggests the meeting brought together workers interested in the interplay between finite and infinite combinatorics, set theory, graph theory and logic. It used to be that infinite set theory, finite combinatorics and logic could be viewed as quite separate and independent subjects. But more and more those disciplines grow together and become interdependent of each other with ever more problems and results appearing which concern all of those disciplines. I appreciate the financial support which was provided by the N. A. T. O. Advanced Study Institute programme, the Natural Sciences and Engineering Research Council of Canada and the Department of Mathematics and Statistics of the University of Calgary. 11l'te meeting on Finite and Infinite Combinatorics in Sets and Logic followed two other meetings on discrete mathematics held in Banff, the Symposium on Ordered Sets in 1981 and the Symposium on Graphs and Order in 1984. The growing inter-relation between the different areas in discrete mathematics is maybe best illustrated by the fact that many of the participants who were present at the previous meetings also attended this meeting on Finite and Infinite Combinatorics in Sets and Logic.

This treatise presents an integrated perspective on the interplay of set theory and graph theory, providing an extensive selection of examples that highlight how methods from one theory can be used to better solve problems originated in the other. Features: explores the interrelationships between sets and graphs and their applications to finite combinatorics; introduces the fundamental graph-theoretical notions from the standpoint of both set theory and dyadic logic, and presents a discussion on set universes; explains how sets can conveniently model graphs, discussing set graphs and set-theoretic representations of claw-free graphs; investigates when it is convenient to represent sets by graphs, covering counting and encoding problems, the random generation of sets, and the analysis of infinite sets; presents excerpts of formal proofs concerning graphs, whose correctness was verified by means of an automated proof-assistant; contains numerous exercises, examples, definitions, problems and insight panels.

"These recent developments also open up new questions of debate, including: What is the role played by randomness? Are computers capable of handling the infinite through unconventional media of computation? How can one approximate efficiently the finite by the infinite, and conversely the infinite by the finite?" "Well-known authors from around the world, many of them architects of the mathematics and computer science for the new century, contribute to this volume. While mathematical in spirit, contributions have many connections with computer science, cognitive science, linguistics, philosophy, physics, biology and semiotics."--Jacket.

In 1992, when Paul Erdos was awarded a Doctor Honoris Causa by Charles University in Prague, a small conference was held, bringing together a distin guished group of researchers with interests spanning a variety of fields related to Erdos' own work. At that gathering, the idea occurred to several of us that it might be quite appropriate at this point in Erdos' career to solicit a col lection of articles illustrating various aspects of Erdos' mathematical life and work. The response to our solicitation was immediate and overwhelming, and these volumes are the result. Regarding the organization, we found it convenient to arrange the papers into six chapters, each mirroring Erdos' holistic approach to mathematics. Our goal was not merely a (random) collection of papers but rather a thor oughly edited volume composed in large part by articles explicitly solicited to illustrate interesting aspects of Erdos and his life and work. Each chap ter includes an introduction which often presents a sample of related Erdos' problems "in his own words". All these (sometimes lengthy) introductions were written jointly by editors. We wish to thank the nearly 70 contributors for their outstanding efforts (and their patience). In particular, we are grateful to Bela Bollobas for his extensive documentation of Paul Erdos' early years and mathematical high points (in the first part of this volume); our other authors are acknowledged in their respective chapters. We also want to thank A. Bondy, G. Hahn, I.

This book has arisen from a colloquium held at St. John's College, Cambridge, in July 1989, which brought together most of today's leading experts in the field of infinite graph theory and combinatorics. This was the first such meeting ever held, and its aim was to assess the state of the art in the discipline, to consider its links with other parts of mathematics, and to discuss possible directions for future development. This volume reflects the Cambridge meeting in both level and scope. It contains research papers as well as expository surveys of particular areas. Together they offer a comprehensive portrait of infinite graph theory and combinatorics, which should be particularly attractive to anyone new to the discipline.