Set‐homogeneous hypergraphs

Abstract

A ‐uniform hypergraph is set‐homogeneous if it is countable (possibly finite) and whenever two finite induced subhypergraphs are isomorphic there is with ; the hypergraph 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 ‐uniform hypergraphs that are not homogeneous (two with , one with and one with ). Evidence is also given that these may be the only ones, up to complementation. For example, for there is just one countably infinite ‐uniform hypergraph whose automorphism group is not 2‐transitive, and there is none for . We also give an example of a finite set‐homogeneous 3‐uniform hypergraph that is not homogeneous.