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Intersecting Systems

  • R. AHLSWEDE (a1), N. ALON (a2), P. L. ERDŐS (a3), M. RUSZINKÓ (a4) and L. A. SZÉKELY (a5)...
    • Published online: 01 June 1997
Abstract

An intersecting system of type (∃, ∀, k, n) is a collection [ ]={[Fscr ]1, ..., [Fscr ]m} of pairwise disjoint families of k-subsets of an n-element set satisfying the following condition. For every ordered pair [Fscr ]i and [Fscr ]j of distinct members of [ ] there exists an A∈[Fscr ]i that intersects every B∈[Fscr ]j. Let In (∃, ∀, k) denote the maximum possible cardinality of an intersecting system of type (∃, ∀, k, n). Ahlswede, Cai and Zhang conjectured that for every k≥1, there exists an n0(k) so that In (∃, ∀, k)=(n−1/k−1) for all n>n0(k). Here we show that this is true for k≤3, but false for all k≥8. We also prove some related results.

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Combinatorics, Probability and Computing
  • ISSN: 0963-5483
  • EISSN: 1469-2163
  • URL: /core/journals/combinatorics-probability-and-computing
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