Skip to main content
SpringerLink
Log in

Degrees of freedom versus dimension for containment orders

  • Published:
Order Aims and scope Submit manuscript

Abstract

Given a family of sets L, where the sets in L admit k ‘degrees of freedom’, we prove that not all (k+1)-dimensional posets are containment posets of sets in L. Our results depend on the following enumerative result of independent interest: Let P(n, k) denote the number of partially ordered sets on n labeled elements of dimension k. We show that log P(n, k)∼nk log n where k is fixed and n is large.

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. N. Alon (1986) The number of polytopes, configurations and real matroids, Mathematika 33, 62–71.

    Google Scholar 

  2. B. Dushnik and E. Miller (1941) Partially ordered sets, Amer. J. Math. 63, 600–610.

    Google Scholar 

  3. P. C. Fishburn and W. T. TrotterJr. (1985) Angle orders, Order 1, 333–343.

    Google Scholar 

  4. M. C. Golumbic and E. R. Scheinerman (1985) Containment graphs, posets and related classes of graphs, Proc. N.Y.A.S., to appear.

  5. A. J. Lehel Gyárfás and Zs. Tuza (1985) How many atoms can be defined by boxes?, Combinatorica 5, 193–204.

    Google Scholar 

  6. T. Hiraguchi (1951) On the dimension of partially ordered sets, Sci. Rep. Kanazawa Univ. 1, 77–94.

    Google Scholar 

  7. N. Santoro and J. Urrutia (1987) Angle orders, regular n-gon orders and the crossing number, Order 4, 209–220.

    Google Scholar 

  8. E. R. Scheinerman and J. C. Wierman (1988) On circle containment orders, Order 4, 315–318.

    Google Scholar 

  9. J. B. Sidney, S. J. Sidney, and J. Urrutia (1988) Circle orders, n-gon orders and the crossing number, Order 5, 1–10.

    Google Scholar 

  10. Warren, H. E. (1968) Lower bounds for approximation by nonlinear manifolds, Trans. A.M.S. 133, 167–178.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Tel Aviv University, 69978, Ramat Aviv, Israel

    Noga Alon

  2. Department of Mathematical Sciences, The Johns Hopkins University, 21218, Baltimore, Maryland, USA

    Edward R. Scheinerman

Authors
  1. Noga Alon
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Edward R. Scheinerman
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by I. Rival

Research supported in part by Allon Fellowship and by a grant from Bat Sheva de Rothschild Foundation.

Research supported in part by the Office of Naval Research, contract number N00014-85-K0622.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Alon, N., Scheinerman, E.R. Degrees of freedom versus dimension for containment orders. Order 5, 11–16 (1988). https://doi.org/10.1007/BF00143892

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00143892

AMS subject classifications (1980)

Key words

Search

Navigation