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, almoet completely, with copies of K_k^r, where k is fixed. This result is considered one of the most fruitful applications of the probabilistic method. It was not known whether the same result also holds in a dense hypergraph setting. In this paper we prove that it does. We prove 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 on edges has an H₀-packing that covers |E(H)|(1 - o_n(1)) edges. Our method of proof is via fractional decompositions. Let H₀ be a fixed hypergraph. A fractional H₀-decomposition of a hypergraph H is an assignment of nonnegative real weights to the copies of H₀ in H such that for each edge e ε E(H), the sum of the weights of copies of H₀ containing e is precisely one. Let k and r be positive integers with k > r > 2. We prove that there exists a constant α = α(k, r) < 1 such that every r-uniform hypergraph with n (sufficiently large) vertices in which every (r - 1)-set is contained in at least on edges has a fractional K_k ^r-decomposition. We then apply a recent result of Rödl, Schacht, Siggers and Tokushige to obtain our integral packing result. The proof makes extensive use of probabilistic arguments and additional combinatorial ideas.