### 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 |

Publisher | Springer Verlag |

Pages | 122-137 |

Number of pages | 16 |

Volume | 8845 |

ISBN (Print) | 9783319132860 |

DOIs | |

Publication status | Published - 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) | 03029743 |

ISSN (Electronic) | 16113349 |

### Fingerprint

### ASJC Scopus subject areas

- Computer Science(all)
- Theoretical Computer Science

### Cite this

*Discrete and Computational Geometry and Graphs - 16th Japanese Conference, JCDCGG 2013, Revised Selected Papers*(Vol. 8845, 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

**Cross-intersecting families of vectors.** / Pach, János; Tardos, G.

Research output: Chapter in Book/Report/Conference proceeding › Chapter

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

}

TY - CHAP

T1 - Cross-intersecting families of vectors

AU - Pach, János

AU - Tardos, G.

PY - 2014

Y1 - 2014

N2 - Given a sequence of positive integers p = (p1,…, pn), let Sp denote the family of all sequences of positive integers x = (x1,…, xn) such that xi ≤ pi for all i. Two families of sequences (or vectors), A, B ⊆ Sp, 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 xi = yi. 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 pi > r + 1. The case min pi ≤ 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.

AB - Given a sequence of positive integers p = (p1,…, pn), let Sp denote the family of all sequences of positive integers x = (x1,…, xn) such that xi ≤ pi for all i. Two families of sequences (or vectors), A, B ⊆ Sp, 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 xi = yi. 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 pi > r + 1. The case min pi ≤ 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.

UR - http://www.scopus.com/inward/record.url?scp=84921362382&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84921362382&partnerID=8YFLogxK

U2 - 10.1007/978-3-319-13287-711

DO - 10.1007/978-3-319-13287-711

M3 - Chapter

AN - SCOPUS:84921362382

SN - 9783319132860

VL - 8845

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 122

EP - 137

BT - Discrete and Computational Geometry and Graphs - 16th Japanese Conference, JCDCGG 2013, Revised Selected Papers

PB - Springer Verlag

ER -