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.

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Proceedings of the NATO Advanced Study Institute, Banff, Alberta, Canada, April 21-May 4, 1991

edited by N.W. Sauer, R.E. Woodrow, B. Sands

Norbert W. Sauer, R. E. Woodrow, B. Sands

Norbert W Sauer; Robert Woodrow; B. Sands

Springer Science + Business Media BV

Softcover reprint of the original 1st ed. 1993, 2012

Springer Nature, Dordrecht, 2012

Netherlands, Netherlands

lg964178

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Front Matter....Pages i-xvii
Extensions of the Erdős—Rado Theorem....Pages 1-17
Zero Sum Trees: A Survey of Results and Open Problems....Pages 19-29
On Superatomic Boolean algebras....Pages 31-62
Representing Rings Using Continuous Functions....Pages 63-79
Homogeneous Directed Graphs....Pages 81-95
Ordinal Partition Behavior of Finite Powers of Cardinals....Pages 97-115
Some Subdirect Products of Finite Nilpotent Groups....Pages 117-124
Three Remarks on End-Faithfulness....Pages 125-133
True Embedding Partition Relations....Pages 135-151
Lattices Related to Separation in Graphs....Pages 153-167
Ramsey Numbers for Sets of Five Vertex Graphs With Fixed Number of Edges....Pages 169-174
Finite Structures with Few Types....Pages 175-187
Recognition Problem in Reconstruction for Decomposable Relations....Pages 189-198
The Average Size of Nonsingular Sets in a Graph....Pages 199-205
Some Canonical Partition Ordinals....Pages 207-224
The Group of Automorphisms of a Relational Saturated Structure....Pages 225-236
On Canonical Ramsey Numbers for Coloring Three-Element Sets....Pages 237-247
Large Subgroups of Infinite Symmetric Groups....Pages 249-278
Homogeneous Partially Ordered Sets....Pages 279-288
Cardinal Representations....Pages 289-298
A Vertex-to-Vertex Pursuit Game Played With Disjoint Sets of Edges....Pages 299-312
Graphs and Posets with no Infinite Independent Set....Pages 313-335
Problems About Planar Orders....Pages 337-347
Superstable and Unstable Theories of Order....Pages 349-353
Advances in Cardinal Arithmetic....Pages 355-383
Conjectures of Rado and Chang and Cardinal Arithmetic....Pages 385-398
Random Structures and Zero-One Laws....Pages 399-420
Isotone Maps: Enumeration and Structure....Pages 421-430
Back Matter....Pages 431-453

This book highlights the newly emerging connections between problems in finite combinatorics and graph theory on the one hand and the more foundational subjects of logic and set theory on the other.
One of the more obvious routes for such a connection is the straightforward generalization of certain definitions and problems from the finite to the infinite, and sometimes the other way around. Another one is to generalize some definitions and find the appropriate new concepts for the new setting, for example the discussion of ends of graphs.
The realization of the importance of homogeneous structures and their connection with logic, as well as with finite structure theory, is a good example of the connection between finite and infinite structures. Almost all of the articles in the present book touch in one way or another on homogeneous structures. The discussion of the 0--1 law, of Ramsey theory for finite and infinite structures, and of divisibility theory highlight this.

2013-08-01