## Abstract

An r-uniform hypergraph is called t-cancellative if for any t + 2 distinct edges A1, . . . ,At,B,C, it holds that (∪ ti =1Ai) ∪ B not = (∪ t i=1Ai) ∪ C. It is called t-union-free if for any two distinct subsets scrA; , scrB , each consisting of at most t edges, it holds that ∪ Ain scrA; A not = ∪ Bin scrB B. Let Ct(n, r) (resp., Ut(n, r)) denote the maximum number of edges of a t-cancellative (resp., t-union-free) r-uniform hypergraph on n vertices. Among other results, we show that for fixed r geq 3, t geq 3 and n rightarrow infty , ω (nlfloor 2r t+2 rfloor; +2r ({m}{o}{d} t+2) t+1 ) = Ct(n, r) = O(n lceil r lfloor t/2⌋ +1 ⌉ ) and Ω (n r t 1 ) = Ut(n, r) = O(nlceil r t 1 ⌉ ), thereby significantly narrowing the gap between the previously known lower and upper bounds. In particular, we determine the Turán exponent of Ct(n, r) when 2 | t and (t/2 + 1) | r, and of Ut(n, r) when (t 1) | r. The main tool used in proving the two lower bounds is a novel connection between these problems and sparse hypergraphs.

Original language | English |
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Pages (from-to) | 2338-2345 |

Number of pages | 8 |

Journal | SIAM Journal on Discrete Mathematics |

Volume | 34 |

Issue number | 4 |

DOIs | |

State | Published - 2020 |

## Keywords

- Cancellative hypergraphs
- Hypergraph Turán-type problems
- Sparse hypergraphs
- Union-free hypergraphs

## All Science Journal Classification (ASJC) codes

- General Mathematics