### Abstract

Given a sequence of positive integers p = (p_{1},…, p_{n}), let Sp denote the family of all sequences of positive integers x = (x_{1},…, x_{n}) such that xi ≤ pi 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, Frankl, Füredi, Livingston, Moon, and Tokushige, and answers a question of Zhang. The special case r = 1 has also been settled recently by Borg.

Original language | English |
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Title of host publication | Discrete and Computational Geometry and Graphs - 16th Japanese Conference, JCDCGG 2013, Revised Selected Papers |

Editors | Hiro Ito, Toshinori Sakai, Jin Akiyama |

Publisher | Springer Verlag |

Pages | 122-137 |

Number of pages | 16 |

ISBN (Electronic) | 9783319132860 |

DOIs | |

Publication status | Published - Jan 1 2014 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 8845 |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

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## Cite this

*Discrete and Computational Geometry and Graphs - 16th Japanese Conference, JCDCGG 2013, Revised Selected Papers*(pp. 122-137). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 8845). Springer Verlag. https://doi.org/10.1007/978-3-319-13287-711