## Abstract

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 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 on edges has an H_{0}-packing that covers |E(H)|(1 - o_{n}(1)) edges. Our method of proof is via fractional decompositions. Let H_{0} be a fixed hypergraph. A fractional H_{0}-decomposition of a hypergraph H is an assignment of nonnegative real weights to the copies of H_{0} in H such that for each edge e ε E(H), the sum of the weights of copies of H_{0} 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.

Original language | English |
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Pages (from-to) | 482-493 |

Number of pages | 12 |

Journal | Lecture Notes in Computer Science |

Volume | 3624 |

DOIs | |

State | Published - 2005 |

Event | 8th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2005 and 9th International Workshop on Randomization and Computation, RANDOM 2005 - Berkeley, CA, United States Duration: 22 Aug 2005 → 24 Aug 2005 |

## ASJC Scopus subject areas

- Theoretical Computer Science
- General Computer Science