Set-homogeneous hypergraphs

Abstract

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 ∈ 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 which 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 which is not homogeneous.