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Saturating Sperner Families

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Abstract

A family \({\mathcal{F} \subseteq 2^{[n]}}\) saturates the monotone decreasing property \({\mathcal{P}}\) if \({\mathcal{F}}\) satisfies \({\mathcal{P}}\) and one cannot add any set to \({\mathcal{F}}\) such that property \({\mathcal{P}}\) is still satisfied by the resulting family. We address the problem of finding the minimum size of a family saturating the k-Sperner property and the minimum size of a family that saturates the Sperner property and that consists only of l-sets and (l + 1)-sets.

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Author notes

  1. Nathan Lemons

    Present address: Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM, 87545, USA

  2. Cory Palmer

    Present address: University of Illinois, Urbana-Champaign, Urbana, IL, 61801, USA

Authors and Affiliations

  1. Hungarian Academy of Sciences, Alfréd Rényi Institute of Mathematics, P.O.B. 127, Budapest, 1364, Hungary

    Dániel Gerbner, Balázs Keszegh, Nathan Lemons, Cory Palmer & Balázs Patkós

  2. Department of Computer Science, Eötvös Loránd University, Pázmány Péter sétány 1/C, Budapest, 1117, Hungary

    Dömötör Pálvölgyi

Authors
  1. Dániel Gerbner
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  2. Balázs Keszegh
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  3. Nathan Lemons
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  4. Cory Palmer
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  5. Dömötör Pálvölgyi
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  6. Balázs Patkós
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Corresponding author

Correspondence to Cory Palmer.

Additional information

The research of D. Gerbner, B. Keszegh and C. Palmer was supported by Hungarian National Scientific Fund, Grant number: OTKA NK-78439.

The European Union and the European Social Fund have provided financial support to the project under the Grant Agreement No. TÁMOP 4.2.1./B-09/1/KMR-2010-0003 to D. Pálvölgyi.

The research of B. Patkós’s was supported by Hungarian National Scientific Fund, Grant Numbers: OTKA K-69062 and PD-83586.

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Gerbner, D., Keszegh, B., Lemons, N. et al. Saturating Sperner Families. Graphs and Combinatorics 29, 1355–1364 (2013). https://doi.org/10.1007/s00373-012-1195-6

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  • DOI: https://doi.org/10.1007/s00373-012-1195-6

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