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Families of sets with locally bounded width

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Abstract

A family of sets ℱ islocally k- wide if and only if the width (as a poset ordered by inclusion) of ℱ is at mostk for everyx. The directed covering graph of a locally 1-wide family of sets is a forest of rooted trees. It is shown that if ℱ is a locallyk-wide family of subsets of {1,...,n}, then |ℱ|≤(2k)k−1 n. The proof involves a counting argument based on families of closed sets associated with theSperner closures in the filters of ℱ. The Sperner closure ofU in ℱ is the intersection of the members of the greatest Sperner antichain of ℋ U = {V ∈ ℋ|V ⊇U}. This closure operation is related to a generalization of maximality in posets.

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References

  1. Dilworth, R.P.: A decomposition theorem for partially ordered sets. Ann. of Math.51, 161–166 (1950)

    Article  MathSciNet  Google Scholar 

  2. Dilworth, R.P.: Some combinatorial problems on partially ordered sets. In Combinatorial Analysis, Proc. Symp. Appl. Math. (American Mathematical Society, Providence, R.I., 1960) 85

  3. Greene, C, Kleitman, DJ.: The structure of Sperner k-families. J. Comb. Theory Ser. A20, 41–68 (1976)

    Article  MathSciNet  Google Scholar 

  4. Grimaldi, R.P.: Discrete and Combinatorial Mathematics, an Applied Introduction, 2nd ed., (Addison-Wesley Pub. Reading, Massachusetts, 1989)

    Google Scholar 

  5. Kendig, K.: Elementary Algebraic Geometry (Springer Verlag, New York, 1977)

    MATH  Google Scholar 

  6. Knill, E.: Generalized degrees and densities for families of sets, Ph.D. Thesis, University of Colorado, Boulder, CO, 1991

    Google Scholar 

  7. Knill, E.: Extreme k-families. Eur. J. Comb. (to be published)

  8. Stanley, R.P.: Enumerative Combinatorics, Vol I, Wadsworth&Brooks/Cole, Monterey, Cal., 1986

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  1. Los Alamos National Laboratory, Mailstop B265, P.O. Box 1663, 87545, Los Alamos, NM

    Emanuel Knill

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  1. Emanuel Knill
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Knill, E. Families of sets with locally bounded width. Graphs and Combinatorics 10, 145–157 (1994). https://doi.org/10.1007/BF02986659

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  • DOI: https://doi.org/10.1007/BF02986659

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