Fractional decompositions of dense hypergraphs

A seminal result of Rödl (the Rödl nibble ) asserts that the edges of the complete r ‐uniform hypergraph K n r can be packed, almost completely, with copies of K k r , 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 H 0 , there exists a constant α = α( H 0 ) < 1 such that every r ‐uniform hypergraph H in which every ( r − 1)‐set is contained in at least α n edges has an H 0 ‐packing that covers | E ( H )|(1 − o n (1)) edges. Our method of proof uses fractional decompositions and makes extensive use of probabilistic arguments and additional combinatorial ideas.