Uniformly cross intersecting families

Noga Alon, Eyal Lubetzky

Research output: Contribution to journalArticle

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(n) the maximum value of {pipe}A{pipe}{pipe}B{pipe} over all such pairs. The best known upper bound on P(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 P(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.

Original languageEnglish (US)
Pages (from-to)389-431
Number of pages43
JournalCombinatorica
Volume29
Issue number4
DOIs
StatePublished - 2009

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Computational Mathematics

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