On the divisibility of homogeneous hypergraphs |
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Authors: | M El-Zahar N Sauer |
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Institution: | (1) Ain Shams University, Cairo, Egypt;(2) Kuwait University, P. O. BOX 5969, Kuwait;(3) University of Calgary, Calgary, Canada |
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Abstract: | We denote byK
k
,k, ![ell](/content/l61223g6h70723lq/xxlarge8467.gif) 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 inK
2
2
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 ![ell](/content/l61223g6h70723lq/xxlarge8467.gif) 3 are indivisible. The -free hypergraphs with K
k
2
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. |
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Keywords: | 04 A 20 05 C 55 |
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