Sharpening the LYM inequality

Abstract

The level sequence of a Sperner familyF is the sequencef(F)={f i (F)}, wheref i (F) is the number ofi element sets ofF . TheLYM inequality gives a necessary condition for an integer sequence to be the level sequence of a Sperner family on ann element set. Here we present an indexed family of inequalities that sharpen theLYM inequality.

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References

  1. [1]

    G. C. Clements: A minimization problem concerning subsets of finite sets,Discrete Mathematics 4 (1973), 123–128.

    Google Scholar 

  2. [2]

    D. E. Daykin, J. Godfrey, andA. J. W. Hilton: Existence theorems for Sperner families.J. Combinatorial Theory (A)17 (1974), 245–251.

    Google Scholar 

  3. [3]

    C. Greene, andA. J. W. Hilton: Some results on Sperner families,J. Comb. Theory (A)26 (1979), 202–209.

    Google Scholar 

  4. [4]

    G. O. H. Katona: A theorem of finite sets,Theory of graphs, Proc. Coll. Tihany, 1966, Akadémiai Kiadó, 1966, 187–207.

  5. [5]

    J. B. Kruskal: The number of simplices in a complex,Mathematical Optimization Techniques, Univ. of Calif. Press, Berkeley and Los Angeles (1963), 251–278.

    Google Scholar 

  6. [6]

    D. J. Kleitman, J. Sha: The number of linear extensions of subset ordering,Discrete Mathematics 63 (1978), 271–278.

    Google Scholar 

  7. [7]

    D. Lubell: A short proof of Sperner's lemma,J. Combinatorial Theory 1 (1966), 299.

    Google Scholar 

  8. [8]

    L. D. Meshalkin: A generalization of Sperner's theorem on the number of subsets of a finite set.Teor. Verojatnost i Primen. 8 (1963), 219–220, in Russian.

    Google Scholar 

  9. [9]

    E. Sperner: Ein Satz über Untermengen einer endliche Menge,Math. Z. 27 (1928), 544–548.

    Google Scholar 

  10. [10]

    K. Yamamoto: Logarithmic order of free distributive lattices.J. Math. Soc. Japan 6 (1954), 343–353.

    Google Scholar 

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Research supported in part by Alexander v. Humboldt-Stiftung

Research supported in part by NSF under grant DMS-86-06225 and AFOSR grant OSR-86-0078

Research supported in part by NSF grant CCR-8911388

Research supported in part by OTKA 327 0113

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Erdős, P.L., Frankl, P., Kleitman, D.J. et al. Sharpening the LYM inequality. Combinatorica 12, 287–293 (1992). https://doi.org/10.1007/BF01285817

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AMS subject classification code (1991)

  • 05 D 05