Maximal antichains of minimum size

Thomas Kalinowski, Uwe Leck, Ian Roberts

    Research output: Contribution to journalArticleResearchpeer-review

    Abstract

    Let n⩾4 be a natural number, and let K be a set K⊆[n]:={1,2,…,n} . We study the problem of finding the smallest possible size of a maximal family A of subsets of [n] such that A contains only sets whose size is in K , and A⊈B for all {A,B}⊆A , i.e. A is an antichain. We present a general construction of such antichains for sets K containing 2, but not 1. If 3∈K our construction asymptotically yields the smallest possible size of such a family, up to an o(n 2 ) error. We conjecture our construction to be asymptotically optimal also for 3∉K , and we prove a weaker bound for the case K={2,4} . Our asymptotic results are straightforward applications of the graph removal lemma to an equivalent reformulation of the problem in extremal graph theory, which is interesting in its own right.
    Original languageEnglish
    Pages (from-to)P3-1-P3-14
    Number of pages14
    JournalElectronic Journal of Combinatorics
    Volume20
    Issue number2
    Publication statusPublished - 2013

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    Antichain
    Extremal Graph Theory
    Graph theory
    Asymptotically Optimal
    Reformulation
    Natural number
    Lemma
    Subset
    Graph in graph theory
    Family

    Cite this

    Kalinowski, T., Leck, U., & Roberts, I. (2013). Maximal antichains of minimum size. Electronic Journal of Combinatorics, 20(2), P3-1-P3-14.
    Kalinowski, Thomas ; Leck, Uwe ; Roberts, Ian. / Maximal antichains of minimum size. In: Electronic Journal of Combinatorics. 2013 ; Vol. 20, No. 2. pp. P3-1-P3-14.
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    title = "Maximal antichains of minimum size",
    abstract = "Let n⩾4 be a natural number, and let K be a set K⊆[n]:={1,2,…,n} . We study the problem of finding the smallest possible size of a maximal family A of subsets of [n] such that A contains only sets whose size is in K , and A⊈B for all {A,B}⊆A , i.e. A is an antichain. We present a general construction of such antichains for sets K containing 2, but not 1. If 3∈K our construction asymptotically yields the smallest possible size of such a family, up to an o(n 2 ) error. We conjecture our construction to be asymptotically optimal also for 3∉K , and we prove a weaker bound for the case K={2,4} . Our asymptotic results are straightforward applications of the graph removal lemma to an equivalent reformulation of the problem in extremal graph theory, which is interesting in its own right.",
    author = "Thomas Kalinowski and Uwe Leck and Ian Roberts",
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    language = "English",
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    pages = "P3--1--P3--14",
    journal = "Electronic Journal of Combinatorics",
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    Kalinowski, T, Leck, U & Roberts, I 2013, 'Maximal antichains of minimum size', Electronic Journal of Combinatorics, vol. 20, no. 2, pp. P3-1-P3-14.

    Maximal antichains of minimum size. / Kalinowski, Thomas; Leck, Uwe; Roberts, Ian.

    In: Electronic Journal of Combinatorics, Vol. 20, No. 2, 2013, p. P3-1-P3-14.

    Research output: Contribution to journalArticleResearchpeer-review

    TY - JOUR

    T1 - Maximal antichains of minimum size

    AU - Kalinowski, Thomas

    AU - Leck, Uwe

    AU - Roberts, Ian

    PY - 2013

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    N2 - Let n⩾4 be a natural number, and let K be a set K⊆[n]:={1,2,…,n} . We study the problem of finding the smallest possible size of a maximal family A of subsets of [n] such that A contains only sets whose size is in K , and A⊈B for all {A,B}⊆A , i.e. A is an antichain. We present a general construction of such antichains for sets K containing 2, but not 1. If 3∈K our construction asymptotically yields the smallest possible size of such a family, up to an o(n 2 ) error. We conjecture our construction to be asymptotically optimal also for 3∉K , and we prove a weaker bound for the case K={2,4} . Our asymptotic results are straightforward applications of the graph removal lemma to an equivalent reformulation of the problem in extremal graph theory, which is interesting in its own right.

    AB - Let n⩾4 be a natural number, and let K be a set K⊆[n]:={1,2,…,n} . We study the problem of finding the smallest possible size of a maximal family A of subsets of [n] such that A contains only sets whose size is in K , and A⊈B for all {A,B}⊆A , i.e. A is an antichain. We present a general construction of such antichains for sets K containing 2, but not 1. If 3∈K our construction asymptotically yields the smallest possible size of such a family, up to an o(n 2 ) error. We conjecture our construction to be asymptotically optimal also for 3∉K , and we prove a weaker bound for the case K={2,4} . Our asymptotic results are straightforward applications of the graph removal lemma to an equivalent reformulation of the problem in extremal graph theory, which is interesting in its own right.

    M3 - Article

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    SP - P3-1-P3-14

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    ER -

    Kalinowski T, Leck U, Roberts I. Maximal antichains of minimum size. Electronic Journal of Combinatorics. 2013;20(2):P3-1-P3-14.