## Abstract

The r-uniform linear k-cycle C_{k} ^{r} is the r-uniform hypergraph on k(r−1) vertices whose edges are sets of r consecutive vertices in a cyclic ordering of the vertex set chosen in such a way that every pair of consecutive edges share exactly one vertex. Here, we prove a balanced supersaturation result for linear cycles which we then use in conjunction with the method of hypergraph containers to show that for any fixed pair of integers r,k≥3, the number of C_{k} ^{r}-free r-uniform hypergraphs on n vertices is 2^{Θ(nr−1)}, thereby settling a conjecture due to Mubayi and Wang from 2017.

Original language | English (US) |
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Pages (from-to) | 309-321 |

Number of pages | 13 |

Journal | Journal of Combinatorial Theory. Series B |

Volume | 134 |

DOIs | |

State | Published - Jan 2019 |

## Keywords

- Asymptotic enumeration
- Balanced supersaturation

## ASJC Scopus subject areas

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