Two problems on homogeneous structures, revisited

Abstract

We take up Peter Cameron's problem of the classification of count- ably infinite graphs which are homogeneous as metric spaces in the graph metric (Cam98). We give an explicit catalog of the known examples, together with results supporting the conjecture that the catalog may be complete, or nearly so. We begin in Part I with a presentation of Fra¨osse's theory of amalgamation classes and the classification of homogeneous structures, with emphasis on the case of homogeneous metric spaces, from the discovery of the Urysohn space to the connection with topological dynamics developed in (KPT05). We then turn to a discussion of the known metrically homogeneous graphs in Part II. This includes a 5-parameter family of homogeneous metric spaces whose connections with topological dynamics remain to be worked out. In the case of diameter 4, we find a variety of examples buried in the tables at the end of (Che98), which we decode and correlate with our catalog. In the final Part we revisit an old chestnut from the theory of homoge- neous structures, namely the problem of approximating the generic triangle free graph by finite graphs. Little is known about this, but we rephrase the problem more explicitly in terms of finite geometries. In that form it leads to questions that seem appropriate for design theorists, as well as some ques- tions that involve structures small enough to be explored computationally. We also show, following a suggestion of Peter Cameron (1996), that while strongly regular graphs provide some interesting examples, one must look beyond this class in general for the desired approximations.