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Incomparable Copies of a Poset in the Boolean Lattice

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Abstract

Let \(\mathcal {B}_{n}\) be the poset generated by the subsets of [n] with the inclusion relation and let \(\mathcal {P}\) be a finite poset. We want to embed \(\mathcal {P}\) into \(\mathcal {B}_{n}\) as many times as possible such that the subsets in different copies are incomparable. The maximum number of such embeddings is asymptotically determined for all finite posets \(\mathcal {P}\) as \(\frac {1}{t(\mathcal {P})}\left (\begin {array}{c}n \\ \lfloor n/2\rfloor \end {array}\right )\), where \(t(\mathcal {P})\) denotes the minimal size of the convex hull of a copy of \(\mathcal {P}\). We discuss both weak and strong (induced) embeddings.

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Authors and Affiliations

  1. MTA Alfréd Rényi Institute of Mathematics, Budapest, Hungary

    Gyula O. H. Katona

  2. Eötvös Loránd University, Budapest, Hungary

    Dániel T. Nagy

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  1. Gyula O. H. Katona
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  2. Dániel T. Nagy
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Correspondence to Gyula O. H. Katona.

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Katona, G.O.H., Nagy, D.T. Incomparable Copies of a Poset in the Boolean Lattice. Order 32, 419–427 (2015). https://doi.org/10.1007/s11083-014-9342-8

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  • DOI: https://doi.org/10.1007/s11083-014-9342-8

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