This is the first of two volumes by Professor Cherlin presenting the state of the art in the classification of homogeneous structures in binary languages and related problems in the intersection of model theory and combinatorics. Researchers and graduate students in the area will find in these volumes many far-reaching results and interesting new research directions to pursue. In this volume, Cherlin develops a complete classification of homogeneous ordered graphs and provides a full proof. He then proposes a new family of metrically homogeneous graphs, a weakening of the usual homogeneity condition. A general classification conjecture is presented, together with general structure theory and applications to a general classification conjecture for such graphs. It also includes introductory chapters giving an overview of the results and methods of both volumes, and an appendix surveying recent developments in the area. An extensive accompanying bibliography of related literature, organized by topic, is available online.
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First of two volumes presenting the state of the art in the classification of binary homogeneous structures and related problems.
1. Results;
2. Methods; Part I. Homogeneous Ordered Graphs:
3. The
catalog of homogeneous ordered graphs;
4. The generically ordered local
order;
5. Ordered homogeneous graphs: Plan of the proof, Propositions IIX;
6. Ordered homogeneous graphs: Proposition I;
7. Ordered homogeneous graphs:
Proposition II;
8. Ordered homogeneous graphs: Proposition III;
9. Ordered
homogeneous graphs: Proposition IV;
10. Ordered homogeneous graphs:
Proposition V; Part II. Metrically Homogeneous Graphs:
11. Metrically
homogeneous graphs: preliminaries;
12. Admissibility allows amalgamation;
13.
Triangle constraints and 4-triviality;
14. Amalgamation requires
admissibility;
15. Local analysis;
16. The bipartite case;
17. Infinite
diameter; Appendix A. Some recent advances; References for Volume I; Index.
Gregory Cherlin is Distinguished Professor Emeritus at Rutgers University. He has worked on applications of model theory to algebra and combinatorics for half a century, and has published four books and over 100 articles on model theory and its applications.
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