On the divisibility of homogeneous hypergraphs

Abstract

We denote byK 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 k then there is exactly one homogeneous hypergraphH whose age is the set of all finite hypergraphs which do not embed any element of ℐ. 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 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 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.

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This research has been supported by NSERC grant 69–1325.

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El-Zahar, M., Sauer, N. On the divisibility of homogeneous hypergraphs. Combinatorica 14, 159–165 (1994). https://doi.org/10.1007/BF01215348

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AMS subject classification code (1991)

  • 04 A 20
  • 05 C 55