### Abstract

A hypergraph (finite set system) ℋ is called a bi-Helly family if it satisfies the following property: if any two edges of a subhypergraph ℋ^{l} ⊆ ℋ share at least two vertices, then |∩^{H∈ℋl}H|≥2. Solving a problem raised by Voloshin, we prove that the maximum number of edges in a bi-Helly family of given order n and given edge size r≥5 equals (^{n-2}_{r-2}). For r = 3 we show that the maximum equals the Turán number ex(n; script K sign^{3}_{4} - e) (its determination is a famous open problem in extremal hypergraph theory), and for r = 4 we prove the lower and upper bounds n^{3}/26 and n^{3}/20, respectively. Analogous results are presented under the requirement that each pairwise k-intersecting subhypergraph has k universal common elements.

Original language | English |
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Pages (from-to) | 321-331 |

Number of pages | 11 |

Journal | Discrete Mathematics |

Volume | 213 |

Issue number | 1-3 |

DOIs | |

Publication status | Published - febr. 28 2000 |

Event | Selected Topics in Discrete Mathematics - Warsaw, Poland Duration: aug. 26 1996 → szept. 28 1996 |

### ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics

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## Cite this

*Discrete Mathematics*,

*213*(1-3), 321-331. https://doi.org/10.1016/S0012-365X(99)00192-2