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Intersecting Systems

Published online by Cambridge University Press:  01 June 1997

R. AHLSWEDE ,
N. ALON ,
P. L. ERDŐS ,
M. RUSZINKÓ  and
L. A. SZÉKELY
R. AHLSWEDE
Affiliation:
Fakultät für Mathematik, Universität Bielefeld, P.O. Box 100131, D-33501 Bielefeld 1, Germany
N. ALON
Affiliation:
Department of Mathematics, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv, Israel (e-mail: [email protected])
P. L. ERDŐS
Affiliation:
Mathematical Institute of the Hungarian Academy of Sciences, Budapest, P.O. Box 127, 1364 Hungary (e-mail: [email protected])
M. RUSZINKÓ
Affiliation:
Computer and Automation Research Institute of the Hungarian Academy of Sciences, Budapest, P.O. Box 63, 1518 Hungary (e-mail: [email protected])
L. A. SZÉKELY
Affiliation:
Department of Computer Science, Eötvös University, Budapest, 1088 Hungary (e-mail: [email protected])
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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.

Type
Research Article
Information
Combinatorics, Probability and Computing , Volume 6 , Issue 2 , June 1997 , pp. 127 - 137
Copyright
1997 Cambridge University Press

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