On the divisibility of homogeneous hypergraphs

Abstract

We denote byK ^ℓ _k ,k, ℓ≥2, the set of allk-uniform hypergraphsK which have the property that every ℓ element subset of the base ofK is a subset of one of the hyperedges ofK. So, the only element inK ²₂ are the complete graphs. If ℐ is a subset ofK ^ℓ _k then there is exactly one homogeneous hypergraphH _ℐ whose age is the set of all finite hypergraphs which do not embed any element of ℐ. We callH _ℐ-free homogeneous graphsH _n have been shown to be indivisible, that is, for any partition ofH _n into two classes, oue of the classes embeds an isomorphic copy ofH _n . [5]. Here we will investigate this question of indivisibility in the more general context ofℐ-free homogeneous hypergraphs. We will derive a general necessary condition for a homogeneous structure to be indivisible and prove that allℐ-free hypergraphs for ℐ ⊂K ^ℓ _k with ℓ≥3 are indivisible. Theℐ-free hypergraphs with ℐ ⊂K ² _k satisfy a weaker form of indivisibility which was first shown by Henson [2] to hold forH _n . The general necessary condition for homogeneous structures to be indivisible will then be used to show that not allℐ-free homogeneous hypergraphs are indivisible.