A Universal Structure for Jv-free Graphs

Countable homogeneous graphs have been classified by Alistair Lachlan and Robert Woodrow [5,11]. There is a countable bipartite graph called the universal 'homogeneous' bipartite graph. However, this graph does not occur in Lachlan and Woodrow's list, because it is not homogeneous as a graph but only as a graph with a fixed bipartition (Cameron [1]). In this paper, I describe a graph which is also very close to being homogeneous, which is universal for the class of N-free graphs. We first describe this class, then analyse why it has no universal homogeneous structure.-This analysis leads us to add a ternary relation to the language of graph theory to obtain a class having a countable universal homogeneous structure, and which has the Af-free graphs as 'underlying structures'. These structures can be shown to be related to classes of trees examined by Peter Cameron [2]. The underlying graph of the resulting homogeneous structure is ubiquitous in category. Cameron has shown that the automorphism group of this graph is a primitive but not doubly transitive Jordan group. Homogeneous structures and their relationship with the Amalgamation Property are discussed in § 1. In § 2, we define the class of AMree graphs and state some of its basic properties. In § 3 we identify the cause of the failure of the Amalgamation Property for the class of N-free graphs. Then in § 4, we define the additional structure required to repair this failure of the Amalgamation Property. Finally, in § 5, we show that there is a countable homogeneous structure whose finite substructures are the structures defined in § 4.