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Optimal pairs of incomparable clouds in multisets

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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, a ≰ b, a ≱ b for all a ∈ A, b ∈ B 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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Authors and Affiliations

  1. Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, 33501, Bielefeld, Germany

    Rudolf Ahlswede

  2. Institute of Problems of Information and Automation, Armenian Academy of Sciences, Erevan-44 P. Sevak str. 1, SFB 343, Visiting, Rep. of Armenia

    Levon H. Khachatrian

Authors
  1. Rudolf Ahlswede
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  2. Levon H. Khachatrian
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Ahlswede, R., Khachatrian, L.H. Optimal pairs of incomparable clouds in multisets. Graphs and Combinatorics 12, 97–137 (1996). https://doi.org/10.1007/BF01858448

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  • DOI: https://doi.org/10.1007/BF01858448

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