Synopses & Reviews
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.
Synopsis
Proceedings of the Advanced Study Institute in Alberta, Canada, in April and May 1991. Topics include extensions of the Erdos-Rado theorem, zero sum trees, superatomic boolean algebras, homogeneous directed graphs, some subdirect products of finite nilpotent groups, true embedding partition relation
Table of Contents
Preface. Extensions of the Erdös--Rado Theorem; J.E. Baumgartner, A. Hajnal, S. Todorcevic. Zero Sum Trees: a Survey of Results and Open Problems; A. Bialostocki. On Superatomic Boolean Algebras; R. Bonnet. Representing Rings using Continuous Functions; A.B. Carson. Homogeneous Directed Graphs; G.L. Cherlin. Ordinal Partition Behavior of Finite Powers of Cardinals; P. Erdoös, A. Hajnal, J.A.A. Larson. Some Subdirect Products of Finite Nilpotent Groups; D.M. Evans. Three Remarks on End-Faithfulness; G. Hahn, J. Sirán. True Embedding Partition Relations; A. Hajnal. Lattices related to Separation in Graphs; R. Halin. Ramsey Numbers for Sets of Five Vertex Graphs with Fixed Number of Edges; H. Harborth. Finite Structures with Few Types; E. Hrushovski. Recognition Problem in Reconstruction for Decomposable Relations; P. Ille. The Average Size of Nonsingular Sets in a Graph; W. Imrich, N. Sauer, W. Woess. Some Canonical Partition Ordinals; J.A. Larson. The Group of Automorphisms of a Relational Saturated Structure; D. Lascar. On Canonical Ramsey Numbers for Coloring Three-Element Sets; H. Lefmann, V. Rödl. Large Subgroups of Infinite Symmetric Groups; D. Macpherson. Homogeneous Partially Ordered Sets; A.H. Mekler. Cardinal Representations; E.C. Milner. A Vertex-to-Vertex Pursuit Game played with Disjoint Sets of Edges; S. Neufeld, R.J. Nowakowski. Graphs and Posets with no Infinite Independent Set; M. Pouzet. Problems about Planar Orders; I. Rival. Superstable and Unstable Theories of Order; T.J. Schneider. Advances in Cardinal Arithmetic; S. Shelah. Conjectures of Rado and Chang and Cardinal Arithmetic; S. Todorčević. Random Structures and Zero--One Laws; P. Winkler. Isotone Maps: Enumeration and Structure; N. Zaguia. Problem Sessions.