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Investigating posets via their maximal chains

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Abstract

By considering the number of maximal chains going through each element of an arbitrary poset, we prove an extension of Erdős’s generalisation of Sperner’s Theorem, together with a partial converse. By considering the number of maximal chains between pairs of comparable elements, we also prove a generalisation of the LYM inequality.

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Acknowledgments

This research was done when the first author was a postdoctoral researcher at IPM (Institute for Research in Fundamental Sciences). He was also supported in part by INSF (Iran National Science Foundation). We would like to thank the anonymous referee, who helped us improve the clarity and accessibility of this paper.

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Correspondence to Seyed Hadi Afzali Borujeni.

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Borujeni, S.H.A., Bowler, N. Investigating posets via their maximal chains. Order (2019) doi:10.1007/s11083-019-09506-2

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Keywords

  • Sperner’s Theorem
  • Maximal chains
  • LYM inequality