## Abstract

We prove that there exists a constant c_{0} such that for any t∈N and any n≥c_{0}t, if A⊂S_{n} is a t-intersecting family of permutations then |A|≤(n−t)!. Furthermore, if |A|≥0.75(n−t)! then there exist i_{1},…,i_{t} and j_{1},…,j_{t} such that σ(i_{1})=j_{1},…,σ(i_{t})=j_{t} holds for any σ∈A. This shows that the conjectures of Deza and Frankl (1977) and of Cameron (1988) on t-intersecting families of permutations hold for all t≤c_{0}n. Our proof method, based on hypercontractivity for global functions, does not use the specific structure of permutations, and applies in general to t-intersecting sub-families of ‘pseudorandom’ families in {1,2,…,n}^{n}, like S_{n}.

Original language | American English |
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Article number | 109650 |

Journal | Advances in Mathematics |

Volume | 445 |

DOIs | |

State | Published - May 2024 |

### Bibliographical note

Publisher Copyright:© 2024 Elsevier Inc.

## Keywords

- Ahlswede-Khachatrian
- Erdős-Ko-Rado
- Forbidden intersection
- Hypercontractivity for global functions
- Intersection problems
- Permutations
- t-intersecting