Skip to main content
SpringerLink
Log in

On Sets of n Points in General Position That Determine Lines That Can Be Pierced by n Points

  • Ricky Pollack Memorial Issue
  • Published:
Discrete & Computational Geometry Aims and scope Submit manuscript

Abstract

Let P be a set of n points in general position in the plane. Let R be a set of n points disjoint from P such that for every \(x,y \in P\) the line through x and y contains a point in R outside of the segment delimited by x and y. We show that \(P \cup R\) must be contained in cubic curve. This resolves a special case of a conjecture of Milićević. We use the same approach to solve a special case of a problem of Karasev related to a bipartite version of the above problem.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  1. Ackerman, E., Buchin, K., Knauer, Ch., Pinchasi, R., Rote, G.: There are not too many magic configurations. Discrete Comput. Geom. 39(1-3), 3–16 (2008)

  2. Blokhuis, A., Marino, G., Mazzocca, F.: Generalized hyperfocused arcs in \(PG(2, p)\). J. Comb. Des. 22(12), 506–513 (2014)

    Article  MathSciNet  Google Scholar 

  3. Brass, P., Moser, W., Pach, J.: Research Problems in Discrete Geometry. Springer, New York (2005)

    MATH  Google Scholar 

  4. Chakerian, G.D.: Sylvester’s problem on collinear points and a relative. Am. Math. Mon. 77, 164–167 (1970)

    Article  MathSciNet  Google Scholar 

  5. Elekes, G.: On linear combinatorics. III. Few directions and distorted lattices. Combinatorica 19(1), 43–53 (1999)

    Article  MathSciNet  Google Scholar 

  6. Erdős, P., Purdy, G.: Some combinatorial problems in the plane. J. Comb. Theory A 25(2), 205–210 (1978)

    Article  MathSciNet  Google Scholar 

  7. Green, B., Tao, T.: On sets defining few ordinary lines. Discrete Comput. Geom. 50(2), 409–468 (2013)

    Article  MathSciNet  Google Scholar 

  8. Grünbaum, B.: Arrangements of colored lines. Notices Am. Math. Soc. 22(1), A-200 (1975)

    Google Scholar 

  9. Jamison, R.E.: A survey of the slope problem. Ann. N. Y. Acad. Sci. 440, 34–51 (1985)

    Article  MathSciNet  Google Scholar 

  10. Jamison, R.E.: Few slopes without collinearity. Discrete Math. 60, 199–206 (1986)

    Article  MathSciNet  Google Scholar 

  11. Karasev, R.: Residues and the combinatorial Nullstellensatz. Period. Math. Hung. 78(2), 157–165 (2019)

    Article  MathSciNet  Google Scholar 

  12. Milićević, L.: Classification theorem for strong triangle blocking arrangements (2018). arXiv:1809.08639

  13. Motzkin, T.S.: Nonmixed connecting lines. Notices Am. Math. Soc. 14(6), 837 (1967)

    Google Scholar 

  14. Murty, U.S.R.: How many magic configurations are there? Am. Math. Mon. 78(9), 1000–1002 (1971)

    Article  MathSciNet  Google Scholar 

  15. Pilatte, C.: On the sets of \(n\) points forming \(n+1\) directions (2020). arXiv:1811.01055

  16. Pinchasi, R.: A solution to a problem of Grünbaum and Motzkin and of Erdős and Purdy about bichromatic configurations of points in the plane. Isr. J. Math. 198(1), 205–214 (2013)

    Article  MathSciNet  Google Scholar 

  17. Pinchasi, R.: Stabbing all lines determined by points in general position in the plane (2018) (manuscript)

  18. Pinchasi, R., Polyanskii, A.: A one-page solution of a problem of Erdős and Purdy (2019) (manuscript)

  19. Scott, P.R.: On the sets of directions determined by \(n\) points. Am. Math. Mon. 77(5), 502–505 (1970)

    Article  MathSciNet  Google Scholar 

  20. Ungar, P.: \(2N\) noncollinear points determine at least \(2N\) directions. J. Comb. Theory A 33(3), 343–347 (1982)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

We thank the anonymous referees for several helpful suggestions.

Author information

Authors and Affiliations

  1. Mathematics Department, Technion – Israel Institute of Technology, 32000, Haifa, Israel

    Chaya Keller & Rom Pinchasi

Authors
  1. Chaya Keller
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Rom Pinchasi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Rom Pinchasi.

Additional information

Editor in Charge: János Pach

Dedicated to the memory of Ricky Pollack.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Chaya Keller and Rom Pinchasi: Research partially supported by Grant 409/16 from the Israel Science Foundation.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Keller, C., Pinchasi, R. On Sets of n Points in General Position That Determine Lines That Can Be Pierced by n Points. Discrete Comput Geom 64, 905–915 (2020). https://doi.org/10.1007/s00454-020-00201-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00454-020-00201-3

Keywords

Search

Navigation