On the divisibility of homogeneous hypergraphs

We denote byK_k^l,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 inK₂² are the complete graphs. If I is a subset ofK_k^l 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_k^l with l=3 are indivisible. TheI-free hypergraphs with I ?K_k² 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.