## Abstract

A k-path is a hypergraph P_{k}={e_{1}, e_{2},..., e_{k}} such that |e_{i}∩e_{j}|=1 if |j-i|=1 and e_{i}∩e_{j}=θ otherwise. A k-cycle is a hypergraph C_{k}={e_{1}, e_{2},..., e_{k}} obtained from a (k-1)-path {e_{1}, e_{2},..., e_{k-1}} by adding an edge e_{k} that shares one vertex with e_{1}, another vertex with e_{k-1} and is disjoint from the other edges. Let ex_{r}(n, G) be the maximum number of edges in an r-graph with n vertices not containing a given r-graph G. We prove that for fixed r≥3, k≥4 and (k, r)≠(4, 3), for large enough n: (equations) if k is odd, if k is even and we characterize all the extremal r-graphs. We also solve the case (k, r)=(4, 3), which needs a special treatment. The case k=3 was settled by Frankl and Füredi.

Original language | English (US) |
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Pages (from-to) | 57-79 |

Number of pages | 23 |

Journal | Journal of Combinatorial Theory. Series A |

Volume | 129 |

DOIs | |

State | Published - Jan 2015 |

## Keywords

- Cycles
- Hypergraph Turán number
- Paths
- Uniform hypergraphs

## ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics