### Abstract

Abstrac: Let A and B denote two families of subsets of an n-element set. The pair (A,B) is said to be ℓ-cross-intersecting iff {pipe}A∩B{pipe} = ℓ for all A ∈ A and B ∈ B. Denote by P_{ℓ}(n) the maximum value of {pipe}A{pipe}{pipe}B{pipe} over all such pairs. The best known upper bound on P_{ℓ}(n) is Θ(2^{n}), by Frankl and Rödl. For a lower bound, Ahlswede, Cai and Zhang showed, for all n ≥ 2ℓ, a simple construction of an ℓ-cross-intersecting pair (A,B) with, and conjectured that this is best possible. Consequently, Sgall asked whether or not P_{ℓ}(n) decreases with ℓ. In this paper, we confirm the above conjecture of Ahlswede et al. for any sufficiently large ℓ, implying a positive answer to the above question of Sgall as well. By analyzing the linear spaces of the characteristic vectors of A, B over ℝ, we show that there exists some ℓ_{0} > 0, such that, for all ℓ ≥ ℓ_{0}. Furthermore, we determine the precise structure of all the pairs of families which attain this maximum.

Original language | English (US) |
---|---|

Pages (from-to) | 389-431 |

Number of pages | 43 |

Journal | Combinatorica |

Volume | 29 |

Issue number | 4 |

DOIs | |

State | Published - 2009 |

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### ASJC Scopus subject areas

- Computational Mathematics
- Discrete Mathematics and Combinatorics

### Cite this

*Combinatorica*,

*29*(4), 389-431. https://doi.org/10.1007/s00493-009-2332-6

**Uniformly cross intersecting families.** / Alon, Noga; Lubetzky, Eyal.

Research output: Contribution to journal › Article

*Combinatorica*, vol. 29, no. 4, pp. 389-431. https://doi.org/10.1007/s00493-009-2332-6

}

TY - JOUR

T1 - Uniformly cross intersecting families

AU - Alon, Noga

AU - Lubetzky, Eyal

PY - 2009

Y1 - 2009

N2 - Abstrac: Let A and B denote two families of subsets of an n-element set. The pair (A,B) is said to be ℓ-cross-intersecting iff {pipe}A∩B{pipe} = ℓ for all A ∈ A and B ∈ B. Denote by Pℓ(n) the maximum value of {pipe}A{pipe}{pipe}B{pipe} over all such pairs. The best known upper bound on Pℓ(n) is Θ(2n), by Frankl and Rödl. For a lower bound, Ahlswede, Cai and Zhang showed, for all n ≥ 2ℓ, a simple construction of an ℓ-cross-intersecting pair (A,B) with, and conjectured that this is best possible. Consequently, Sgall asked whether or not Pℓ(n) decreases with ℓ. In this paper, we confirm the above conjecture of Ahlswede et al. for any sufficiently large ℓ, implying a positive answer to the above question of Sgall as well. By analyzing the linear spaces of the characteristic vectors of A, B over ℝ, we show that there exists some ℓ0 > 0, such that, for all ℓ ≥ ℓ0. Furthermore, we determine the precise structure of all the pairs of families which attain this maximum.

AB - Abstrac: Let A and B denote two families of subsets of an n-element set. The pair (A,B) is said to be ℓ-cross-intersecting iff {pipe}A∩B{pipe} = ℓ for all A ∈ A and B ∈ B. Denote by Pℓ(n) the maximum value of {pipe}A{pipe}{pipe}B{pipe} over all such pairs. The best known upper bound on Pℓ(n) is Θ(2n), by Frankl and Rödl. For a lower bound, Ahlswede, Cai and Zhang showed, for all n ≥ 2ℓ, a simple construction of an ℓ-cross-intersecting pair (A,B) with, and conjectured that this is best possible. Consequently, Sgall asked whether or not Pℓ(n) decreases with ℓ. In this paper, we confirm the above conjecture of Ahlswede et al. for any sufficiently large ℓ, implying a positive answer to the above question of Sgall as well. By analyzing the linear spaces of the characteristic vectors of A, B over ℝ, we show that there exists some ℓ0 > 0, such that, for all ℓ ≥ ℓ0. Furthermore, we determine the precise structure of all the pairs of families which attain this maximum.

UR - http://www.scopus.com/inward/record.url?scp=77955284900&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77955284900&partnerID=8YFLogxK

U2 - 10.1007/s00493-009-2332-6

DO - 10.1007/s00493-009-2332-6

M3 - Article

VL - 29

SP - 389

EP - 431

JO - Combinatorica

JF - Combinatorica

SN - 0209-9683

IS - 4

ER -