On the divisibility of homogeneous hypergraphs

We denote byK_k,k, 2, the set of allk-uniform hypergraphsK which have the property that every 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 is a subset ofK_k then there is exactly one homogeneous hypergraphH whose age is the set of all finite hypergraphs which do not embed any element of . We callH-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 of-free homogeneous hypergraphs. We will derive a general necessary condition for a homogeneous structure to be indivisible and prove that all-free hypergraphs for K_k with 3 are indivisible. The-free hypergraphs with 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 all-free homogeneous hypergraphs are indivisible.This research has been supported by NSERC grant 69–1325.