Partitioning the Boolean lattice into a minimal number of chains of relatively uniform size

Tim Hsu, Mark J. Logan, Shahriar Shahriari, Christopher Towse

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    7 Scopus citations

    Abstract

    Let 2[n] denote the Boolean lattice of order n, that is, the poset of subsets of {1,...,n} ordered by inclusion. Extending our previous work on a question of Füredi, we show that for any c>1, there exist functions e(n)∼ n/2 and f(n)∼c nlogn and an integer N (depending only on c) such that for all n>N, there is a chain decomposition of the Boolean lattice 2[n] into (⌊n/2⌋ n chains, all of which have size between e(n) and f(n). (A positive answer to Füredi's question would imply that the same result holds for some e(n)∼ π/2 n and f(n)=e(n)+1.) The main tool used is an apparently new observation about rank-collection in normalized matching (LYM) posets.

    Original languageEnglish (US)
    Pages (from-to)219-228
    Number of pages10
    JournalEuropean Journal of Combinatorics
    Volume24
    Issue number2
    DOIs
    StatePublished - Feb 2003

    Keywords

    • Boolean lattice
    • Chain decompositions
    • Füredi's problem
    • LYM property
    • Normalized matching property

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