These two volumes by Professor Cherlin present 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 Volume I, the homogeneous ordered graphs are classified, a new family of metrically homogeneous graphs is constructed, and a general classification conjecture is presented, together with general structure theory and applications to a general classification conjecture for such graphs. Volume II continues the analysis into more general expansions of graphs or tournaments by an additional binary relation, called 3-multi-graphs or 3-multi-tournaments, applying and extending the results of Volume I, resulting in a detailed catalogue of such structures and a second classification conjecture. Appendices to both volumes explore recent developments and open questions.
Aimed at graduate students and researchers in combinatorics and model theory, these two volumes provide a systematic exposition of methods used to classify homogeneous structures and the associated structural Ramsey theory of ordered graphs, graphs equipped with their natural metric structure, and similar structures.
