Fractional Decompositions of Dense Hypergraphs

A seminal result of Rödl (the Rödl nibble) asserts that the edges of the complete r-uniform hypergraph Kr n can be packed, almost completely, with copies of Kr k , 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 α = α(H0) < 1 such that every r-uniform hypergraph H in which every (r − 1)-set is contained in at least αn 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.