Set-homogeneous hypergraphs

A k-uniform hypergraph M is set-homogeneous if it is countable (possibly finite) and whenever two finite induced subhypergraphs U,V are isomorphic there is g is an element of Aut(M) with U-g = V; the hypergraph M is said to be homogeneous if in addition every isomorphism between finite induced subhypergraphs extends to an automorphism. We give four examples of countably infinite set-homogeneous k-uniform hypergraphs that are not homogeneous (two with k=3, one with k=4 and one with k=6). Evidence is also given that these may be the only ones, up to complementation. For example, for k=3 there is just one countably infinite k-uniform hypergraph whose automorphism group is not 2-transitive, and there is none for k=4. We also give an example of a finite set-homogeneous 3-uniform hypergraph that is not homogeneous.