## Abstract

Numerous problems in extremal hypergraph theory ask to determine the maximal size of a k-uniform hypergraph on n vertices that does not contain an ‘enlarged’ copy H^{+} of a fixed hypergraph H. These include well-known problems such as the Erdős-Sós ‘forbidding one intersection’ problem and the Frankl-Füredi ‘special simplex’ problem. We present a general approach to such problems, using a ‘junta approximation method’ that originates from analysis of Boolean functions. We prove that any H^{+}-free hypergraph is essentially contained in a ‘junta’ – a hypergraph determined by a small number of vertices – that is also H^{+}-free, which effectively reduces the extremal problem to an easier problem on juntas. Using this approach, we obtain, for all C<k<n/C, a solution of the extremal problem for a large class of H's, which includes the aforementioned problems, and solves them for a large new set of parameters. We apply our method also to the 1974 Erdős-Chvátal simplex conjecture, which asserts that for any [Formula presented], the maximal size of a k-uniform family that does not contain a d-simplex (i.e., d+1 sets with empty intersection such that any d of them intersect) is (n−1k−1). We prove the conjecture for all d and k, provided n>n_{0}(d).

Original language | English |
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Article number | 107991 |

Journal | Advances in Mathematics |

Volume | 392 |

DOIs | |

State | Published - 3 Dec 2021 |

### Bibliographical note

Publisher Copyright:© 2021 Elsevier Inc.

### Funding

Research supported by the Israel Science Foundation (grants no. 402/13 and 1612/17) and by the Binational US-Israel Science Foundation (grant no. 2014290).Research supported by the Adams Fellowship Program of the Israel Academy of Sciences and Humanities.

Funders | Funder number |
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Israel Academy of Sciences and Humanities | |

Israel Science Foundation | 2014290, 402/13, 1612/17 |

## Keywords

- Discrete Fourier analysis
- Erdős-Chvátal simplex conjecture
- Erdős-Ko-Rado theorem
- Extremal combinatorics
- Intersection theorems
- Junta method