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Bulletin of the London Mathematical Society Advance Access originally published online on December 21, 2006
Bulletin of the London Mathematical Society 2007 39(1):156-166; doi:10.1112/blms/bdl013
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© 2006 London Mathematical Society

Fractional decompositions of dense hypergraphs

Raphael Yuster

Department of Mathematics
University of Haifa
Haifa 31905
Israel
raphy{at}math.haifa.ac.il

Received 13 October 2005. Revision received 21 February 2006.

A seminal result of Rödl (the Rödl nibble) asserts that the edges of the complete r-uniform hypergraph Formula can be packed, almost completely, with copies of Formula , where k is fixed. We prove that the same result holds in a dense hypergraph setting. It is shown that for every r-uniform hypergraph H0, there exists a constant {alpha} = {alpha}(H0) < 1 such that every r-uniform hypergraph H in which every (r – 1)-set is contained in at least {alpha} n edges has an H0-packing that covers |E(H)|(1 – on(1)) edges. Our method of proof uses fractional decompositions and makes extensive use of probabilistic arguments and additional combinatorial ideas.


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