Set-homogeneous hypergraphs

TL;DR

This paper explores countably infinite set-homogeneous hypergraphs, providing examples that are not homogeneous and analyzing their automorphism groups, contributing to the classification of such structures.

Contribution

The paper presents four new examples of countably infinite set-homogeneous hypergraphs that are not homogeneous, and investigates their automorphism groups and uniqueness.

Findings

Four examples of non-homogeneous set-homogeneous hypergraphs are provided.

Evidence suggests these may be the only such hypergraphs up to complementation.

For k=3, only one hypergraph with non-2-transitive automorphism group exists; none for k=4.

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 $g\in Aut(M)$ 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.