Skip to main content
SpringerLink
Log in

About the ratio of the size of a maximum antichain to the size of a maximum level in finite partially ordered sets

  • Published:
Combinatorica Aims and scope Submit manuscript

Abstract

LetP be a finite partially ordered set. The lengthl(x) of an elementx ofP is defined by the maximal number of elements, which lie in a chain withx at the top, reduced by one. Letw(P) (d(P)) be the maximal number of elements ofP which have the same length (which form an antichain). Further let\(p^n : = \underbrace {PX...XP}_{n - times}\). The numbers\(r_k : = \mathop {\max }\limits_{P:|P| = k} \frac{{d(P)}}{{w(P)}}\) and\(s_k : = \mathop {\max }\limits_{P:|P| = k} \mathop {\lim }\limits_{n \to \infty } \frac{{d(P^n )}}{{w(P^n )}}\) as well as all partially ordered sets for which these maxima are attained are determined.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. V. B. Alekseev, O čisle monotonnychk-značnych funkcij.Problemy Kibernetiki,28 (1974), 5–24.

    Google Scholar 

  2. K. Engel, Optimal representations, LYM posets, Peck posets, and the Ahlswede—Daykin inequality,Rostock. Math. Kolloq.,26 (1984), 63–68.

    MATH  MathSciNet  Google Scholar 

  3. K. Engel andH.-D. O. F. Gronau, Sperner theory in partially ordered sets,BSB B. G. Teubner Verlagsgesellschaft, Leipzig, (1985),to appear.

  4. K. Engel andN. N. Kuzjurin, An asymptotic formula for the maximum size of anh-family in products of partially ordered sets,J. Combin. Theory Ser. A,37 (1984), 337–347.

    Article  MATH  MathSciNet  Google Scholar 

  5. C. Greene andD. J. Kleitman, Proof techniques in the theory of finite sets,in: Studies in Combinatorics (G.-C. Rota, ed.), MAA Studies in Math. 17,Washington, D. C., (1978), 22–79.

  6. J. R. Griggs, The Sperner property,in: Ordres: Description et Roles (Proc. Lyon 1982, M. Pouzet, ed.), Annals of Discrete Math., (1984), 371–387.

  7. V. K. Korobkov, Nekotorye obobščenia zadači „rasšifrovki“ monotonnych funkcij algebry logiki,Diskret. Analiz,5 (1965), 19–25.

    MATH  MathSciNet  Google Scholar 

  8. V. V. Petrov,Sums of independent random variables, Akademie-Verlag, Berlin, 1975.

    Google Scholar 

  9. D. B. West, Extremal problems in partially ordered sets,in: Ordered Sets (Proc. Banff 1981, I. Rival, ed.), D. Reidel, Dordrecht, (1982), 473–521.

Download references

Author information

Authors and Affiliations

  1. Sektion Mathematik. Wilhelm-Pieck-Universität, 2500, Rostock, German Democratic Republic

    Konrad Engel & Nikolaj Nikolajevič Kuzjurin

Authors
  1. Konrad Engel
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nikolaj Nikolajevič Kuzjurin
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Engel, K., Kuzjurin, N.N. About the ratio of the size of a maximum antichain to the size of a maximum level in finite partially ordered sets. Combinatorica 5, 301–309 (1985). https://doi.org/10.1007/BF02579244

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02579244

AMS subject classification (1980)

Search

Navigation