### Abstract

An intersecting system of type (∃, ∀, k, n) is a collection double-struck F sign = {ℱ_{1},...,ℱ_{m}} of pairwise disjoint families of k-subsets of an n-element set satisfying the following condition. For every ordered pair ℱ_{i} and ℱ_{j} of distinct members of double-struck F sign there exists an A ∈ ℱ_{i} that intersects every B ∈ ℱ_{j}. Let I_{n}(∃, ∀, k) denote the maximum possible cardinality of an intersecting system of type (∃, ∀, k, n). Ahlswede, Cai and Zhang conjectured that for every k ≥ 1, there exists an n_{0}(k) so that I_{n}(∃, ∀, k) = (^{n-1}_{k-1}) for all n > n_{0}(k). Here we show that this is true for k ≤ 3, but false for all k ≥ 8. We also prove some related results.

Original language | English (US) |
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Pages (from-to) | 127-137 |

Number of pages | 11 |

Journal | Combinatorics Probability and Computing |

Volume | 6 |

Issue number | 2 |

DOIs | |

State | Published - Jan 1 1997 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Statistics and Probability
- Computational Theory and Mathematics
- Applied Mathematics

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## Cite this

*Combinatorics Probability and Computing*,

*6*(2), 127-137. https://doi.org/10.1017/S0963548397003003