Graphs and Combinatorics

, Volume 12, Issue 2, pp 97–137 | Cite as

Optimal pairs of incomparable clouds in multisets

  • Rudolf Ahlswede
  • Levon H. Khachatrian
Article
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Abstract

We consider the partially ordered set ([k] n , ≤), which is defined asn-th product of the chain [k] = {0, 1, 2,...,k − 1}, and study pairs (A, B) of incomparable sets A, B ⊂ [k] n , that is, ab, ab for all aA, bB or (in short notation)
.
We are concerned with the growth of the functionsf n : {0, 1,...,k n } → {0, 1,...,k n },n ∈ ℕ, defined byf n (α) = max {|B|: A, B ⊂ [k] n with |A| = α and
} and a characterisation of pairs (A, B), which assume this bound.

In the previously studied casek = 2 our results are considerably sharper than earlier results by Seymour, Hilton, Ahlswede and Zhang.

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Copyright information

© Springer-Verlag 1996

Authors and Affiliations

  • Rudolf Ahlswede
    • 1
  • Levon H. Khachatrian
    • 2
  1. 1.Fakultät für MathematikUniversität BielefeldBielefeldGermany
  2. 2.Institute of Problems of Information and AutomationArmenian Academy of SciencesVisitingRep. of Armenia

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