Classification of Countable Homogeneous 2-Graphs

We classify certain families of homogeneous 2-graphs and prove some results that apply to families of 2-graphs that we have not completely classified.

We classify homogeneous 2-coloured 2-graphs where one component is a disjoint union of complete graphs and the other is the random graph or the generic Kr-free graph for some r. We show that any non-trivial examples are derived from a homogeneous 2-coloured 2-graph where one component is the complete graph and the other is the random graph or
the generic Kr-free graph for some r; and these are in turn either generic or equivalent to one that minimally omits precisely one monochromatic colour-1 (K1,Kt) 2-graph for some t < r.

We also classify homogeneous 2-coloured 2-graphs G where both components are isomorphic and each is either the random graph or the generic K3-free graph; in both cases show that there is an antichain A of monochromatic colour-1 2-graphs all of the form (Ks,Kt) (for some s and t) such that G is equivalent to the homogeneous 2-coloured 2-graph with the specified components that is generic subject to minimally omitting the elements of A.