Convexly independent sets

Abstract

A family of pairwise disjoint compact convex sets is called convexly independent, if none of its members is contained in the convex hull of the union of the other members of the family. The main result of the paper gives an upper bound for the maximum cardinalityh(k, n) of a family ℱ of mutually disjoint compact convex sets such that any subfamily of at mostk members of ℱ is convexly independent, but no subfamily of sizen is.

This is a preview of subscription content, access via your institution.

References

  1. [1]

    T. Bisztriczky, andG. Fejes Tóth: A generalization of the Erdős-Szekeres convexn-gon theorem,J. reine angew. Math. 395(1989) 167–170.

    Google Scholar 

  2. [2]

    T. Bisztriczky, andG. Fejes Tóth: Nine convex sets determine a pentagon with convex sets as vertices,Geometriae Dedicata 31(1989) 89–104.

    Google Scholar 

  3. [3]

    P. ErdőS, andG. Szekeres: A combinatorical problem in geometry,Comp. Math.,2 (1935), 463–470.

    Google Scholar 

  4. [4]

    P. Erdős, andG. Szekeres: On some extremum problems in elementary geometry,Ann. Univ. Sci. Budapest, Eötvös Sect. Math.,3–4 (1960–61), 53–62.

    Google Scholar 

  5. [5]

    R. L. Graham, B. L. Rothschild, andJ. H. Spencer:Ramsey theory, Wiley, New York (1980).

    Google Scholar 

Download references

Author information

Affiliations

Authors

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Bisztriczky, T., Tóth, G.F. Convexly independent sets. Combinatorica 10, 195–202 (1990). https://doi.org/10.1007/BF02123010

Download citation

AMS subject classification (1980)

  • 52 A 10
  • 05 A 17