Fractional decompositions of dense hypergraphs

• Authors: Raphael Yuster
• Venue: APPROX'05/RANDOM'05 Proceedings of the 8th international workshop on Approximation, Randomization and Combinatorial Optimization Problems, and Proceedings of the 9th international conference on Randamization and Computation: algorithms and techniques
• Year: 2005

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. 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) 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 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 ∈ 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) r-uniform hypergraph with n (sufficiently large) vertices in which every (r–1)-set is contained in at least αn 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.