### Abstract

A k-uniform linear path of length ℓ, denoted by ℙ_{ℓ}^{(k)}, is a family of k-sets {F_{1},...,F_{ℓ}such that {pipe}F_{i}∩ F_{i+1}{pipe}=1 for each i and F_{i}∩ F_{j} = ∅ whenever {pipe}i-j{pipe}>1. Given a k-uniform hypergraph H and a positive integer n, the k-uniform hypergraph Turán number of H, denoted by ex_{k}(n, H), is the maximum number of edges in a k-uniform hypergraph F on n vertices that does not contain H as a subhypergraph. With an intensive use of the delta-system method, we determine ex_{k}(n, P_{ℓ}^{(k)}exactly for all fixed ℓ ≥1, k≥4, and sufficiently large n. We show that (Formula presented.). The only extremal family consists of all the k-sets in [n] that meet some fixed set of t vertices. We also show that (Formula presented.), and describe the unique extremal family. Stability results on these bounds and some related results are also established.

Original language | English |
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Pages (from-to) | 299-322 |

Number of pages | 24 |

Journal | Combinatorica |

Volume | 34 |

Issue number | 3 |

DOIs | |

Publication status | Published - Jun 2014 |

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Computational Mathematics

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## Cite this

*Combinatorica*,

*34*(3), 299-322. https://doi.org/10.1007/s00493-014-2838-4