Infinite imprimitive homogeneous 3-edge-colored complete graphs

Abstract

Introduction. We will refer to a complete graph with a 3-edge coloring as a 3-graph in the present paper, for lack of a better name; equivalently, a 3-graph is a structure in a language consisting of three nontrivial symmetric 2-types. A 3-graph is said to be homogeneous if every isomorphism between finite induced 3-subgraphs extends to an automorphism of the whole 3-graph. The finite homogeneous 3-graphs are known [10, 6], and in fact there is a decent theory of finite homogeneous structures for relational languages [5, 9] which provides a very rough classification in general. The present paper classifies the infinite imprimitive homogeneous 3-graphs, as part of an ongoing project to explore the ramifications of a combinatorial method due primarily to Lachlan. There is no systematic theory of infinite homogeneous structures, even for binary languages. It is not clear whether the infinite homogeneous 3-graphs are classifiable. The classification of the infinite homogeneous directed graphs is known, and the case of 3-graphs appears harder but in some ways similar. We expect that an attempt to apply Lachlan's method to the classification of all infinite homogeneous 3-graphs will either lead to a considerable development of that method, or to some completely unfamiliar phenomena. The case treated here is one of the obvious special cases that needs to be dealt with first. The classification of imprimitive homogeneous directed graphs was dealt with in a similar spirit in [1]. In ? 1 we will present a catalog of the examples found with a brief discussion of their properties. With one exception, the five families listed are analogs of imprimitive directed graphs; the exceptional family is the class of product 3-graphs. Homogeneous product structures exist quite generally but are not encountered in the context of directed graphs, as the type structure is too limited. We will see that the first three families in our catalog are rather special in character, while the last two families are of more or less generic type. The bulk of the analysis is directed toward identification theorems for the structures lying in these last two families. In particular the fifth and last family has only two members, each of which needs its own identification theorem. As a result, about a third of our analysis will be devoted to these two structures. The paper is organized as follows. In ? 1 we describe our five families of infinite imprimitive homogeneous 3-graphs, which we refer to as: composite 3-graphs; product 3-graphs; double covers; restricted generic; and generic type. The first four families contain infinitely many examples, while the last family consists of just two