TY - JOUR

T1 - Uniformly cross intersecting families

AU - Alon, Noga

AU - Lubetzky, Eyal

N1 - Funding Information:
* Research supported in part by the Israel Science Foundation, by a USA–Israel BSF grant, by an ERC advanced grant, by NSF grant CCF 0832797 and by the Ambrose Monell Foundation. † Research partially supported by a Charles Clore Foundation Fellowship.

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.

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U2 - 10.1007/s00493-009-2332-6

DO - 10.1007/s00493-009-2332-6

M3 - Article

AN - SCOPUS:77955284900

SN - 0209-9683

VL - 29

SP - 389

EP - 431

JO - Combinatorica

JF - Combinatorica

IS - 4

ER -