Journal of Geometry

, Volume 108, Issue 1, pp 33–43 | Cite as

General position subsets and independent hyperplanes in d-space

Article
First Online:
Received:

Abstract

Erdős asked what is the maximum number \({\alpha(n)}\) such that every set of \({n}\) points in the plane with no four on a line contains \({\alpha(n)}\) points in general position. We consider variants of this question for \({d}\)-dimensional point sets and generalize previously known bounds. In particular, we prove the following two results for fixed \({d}\):

  • Every set \({\mathcal{H}}\) of \({n}\) hyperplanes in \({\mathbb{R}^d}\) contains a subset \({S\subseteq \mathcal{H}}\) of size at least \({c \left(n \log n\right)^{1/d}}\), for some constant \({c=c(d)> 0}\), such that no cell of the arrangement of \({\mathcal{H}}\) is bounded by hyperplanes of \({S}\) only.

  • Every set of \({cq^d\log q}\) points in \({\mathbb{R}^d}\), for some constant \({c=c(d)> 0}\), contains a subset of \({q}\) cohyperplanar points or \({q}\) points in general position.

Two-dimensional versions of the above results were respectively proved by Ackerman et al. [Electronic J. Combinatorics, 2014] and by Payne and Wood [SIAM J. Discrete Math., 2013].

Mathematics Subject Classification

Primary 52C35 Secondary 52C10 
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ackerman, E., Pach, J., Pinchasi, R., Radoičić, R., Tóth, G.: A note on coloring line arrangements. Electron. J. Comb. 21(2), #P2.23 (2014)Google Scholar
  2. 2.
    Bose P., Cardinal J., Collette S., Hurtado F., Korman M., Langerman S., Taslakian P.: Coloring and guarding arrangements. Discret. Math. Theor. Comput. Sci. 15(3), 139–154 (2013)MathSciNetMATHGoogle Scholar
  3. 3.
    Cardinal, J., Felsner, S.: Covering partial cubes with zones. In: Proceedings of the 16th Japan conference on discrete and computational geometry and graphs (JCDCG\({^2}\) 2013), Lecture notes in computer science. Springer, Berlin (2014)Google Scholar
  4. 4.
    Elekes, G., Tóth, C.D.: Incidences of not-too-degenerate hyperplanes. In: Proceedings of the ACM symposium on computational geometry (SoCG), pp. 16–21 (2005)Google Scholar
  5. 5.
    Erdős P.: On some metric and combinatorial geometric problems. Discret. Math. 60, 147–153 (1986)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Füredi Z.: Maximal independent subsets in Steiner systems and in planar sets. SIAM J. Discret. Math. 4(2), 196–199 (1991)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Furstenberg H., Katznelson Y.: A density version of the Hales–Jewett theorem for \({k=3}\). Discret. Math. 75(1-3), 227–241 (1989)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Furstenberg H., Katznelson Y.: A density version of the Hales–Jewett theorem. J. d’analyse Math. 57, 64–119 (1991)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Gowers, T.: A geometric Ramsey problem. http://mathoverflow.net/questions/50928/a-geometric-ramsey-problem (2012)
  10. 10.
    Halperin D.: Arrangements. In: Goodman, J.E., O’Rourke, J. (eds) Handbook of Discrete and Computational Geometry, chapter 24., 2nd edn., CRC Press, Boca Raton (2004)Google Scholar
  11. 11.
    Kostochka A.V., Mubayi D., Verstraëte J.: On independent sets in hypergraphs. Random Struct. Algorithms 44(2), 224–239 (2014)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Milićević, L.: Sets in almost general position (2015). arXiv:1601.07206
  13. 13.
    Payne M.S., Wood D.R.: On the general position subset selection problem. SIAM J. Discret. Math. 27(4), 1727–1733 (2013)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Polymath D.H.J.: A new proof of the density Hales–Jewett theorem. Ann. Math. 175(2), 1283–1327 (2012)MathSciNetMATHGoogle Scholar
  15. 15.
    Solymosi J., Stojaković M.: Many collinear \({k}\)-tuples with no \({k+1}\) collinear points. Discret. Comput. Geom. 50, 811–820 (2013)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Spencer J.: Turán’s theorem for \({k}\)-graphs. Discret. Math. 2(2), 183–186 (1972)CrossRefMATHGoogle Scholar
  17. 17.
    Szemerédi E., Trotter W.T.: Extremal problems in discrete geometry. Combinatorica 3(3), 381–392 (1983)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  • Jean Cardinal
    • 1
  • Csaba D. Tóth
    • 2
  • David R. Wood
    • 3
  1. 1.Université libre de Bruxelles (ULB)BrusselsBelgium
  2. 2.California State University NorthridgeNorthridgeUSA
  3. 3.Monash UniversityMelbourneAustralia

Personalised recommendations