### Abstract

Given a sequence of positive integers p=(p_{1},…,p_{n}), let S_{p} denote the family of all sequences of positive integers x=(x_{1},…,x_{n}) such that x_{i}≤p_{i} for all i. Two families of sequences (or vectors), A, B⊆S_{p}, are said to be r-cross-intersecting if no matter how we select x∈A and y∈B, there are at least r distinct indices i such that x_{i}=y_{i}. We determine the maximum value of |A|·|B| over all pairs of r-cross-intersecting families and characterize the extremal pairs for r≥1, provided that min p_{i}>r+1. The case min p_{i}≤r+1 is quite different. For this case, we have a conjecture, which we can verify under additional assumptions. Our results generalize and strengthen several previous results by Berge, Borg, Frankl, Füredi, Livingston, Moon, and Tokushige, and answers a question of Zhang.

Original language | English |
---|---|

Pages (from-to) | 477-495 |

Number of pages | 19 |

Journal | Graphs and Combinatorics |

Volume | 31 |

Issue number | 2 |

DOIs | |

Publication status | Published - Mar 1 2015 |

### Fingerprint

### ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics

### Cite this

*Graphs and Combinatorics*,

*31*(2), 477-495. https://doi.org/10.1007/s00373-015-1551-4