Two Problems on Homogeneous Structures, Revisited

We take up Peter Cameron’s problem of the classification of countably infinite graphs which are homogeneous as metric spaces in the graph metric [Cam98], working toward an explicit catalog of “known” examples on the one hand, and an investigation of the cases which occur as exceptional cases from the perspective of the catalog. We begin with a presentation of Fräıssé’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 case of metrically homogeneous graphs. We also take this opportunity to revisit another old chestnut from the theory of homogeneous structures, namely the problem of approximating the generic triangle free graph by finite graphs. Very little is known about this, but it is possible to rephrase the problem in somewhat more explicit geometric terms. And in that form one can raise questions that seem appropriate for design theorists, as well as some questions that involve structures small enough to be explored computationally.