Set-homogeneous directed graphs

A directed graph is set-homogeneous if, whenever U and V are isomorphic finite subdigraphs, there is an automorphism g of the digraph with Ug = V . Here, extending work of Lachlan on finite homogeneous digraphs, we classify finite set-homogeneous digraphs, where we allow some pairs of vertices to have arcs in both directions. Under the assumption that such pairs of vertices are not allowed, we obtain initial results on countably infinite set-homogeneous digraphs, classifying those which are not 2-homogeneous.