Skip to main content
SpringerLink
Log in

A One-Page Solution of a Problem of Erdős and Purdy

  • Branko Grünbaum Memorial Issue
  • Published:
Discrete & Computational Geometry Aims and scope Submit manuscript

Abstract

The following theorem was conjectured by Erdős and Purdy: Let P be a set of \(n>4\) points in general position in the plane. Suppose that R is a set of points disjoint from P such that every line determined by P passes through a point in R. Then \(|R| \ge n\). In this paper we give a very elegant and elementary proof of this, being a very good candidate for the “book proof” of this conjecture.

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

Similar content being viewed by others

References

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  5. Grünbaum, B.: Arrangements of colored lines. Abstract 720–50-5. Notices Am. Math. Soc. 22, 200 (1975)

    Google Scholar 

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

  7. Motzkin, T.S.: Nonmixed connecting lines. Abstract 67T 605. Notices Am. Math. Soc. 14, 837 (1967)

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  9. 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 

  10. Pinchasi, R.: Stabbing all lines determined by points in general position in the plane. Eur. J. Combin. (accepted)

Download references

Acknowledgements

We thank Gábor Tardos for a clever remark that helped to shorten the presentation of the proof.

Author information

Authors and Affiliations

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

    Rom Pinchasi

  2. Laboratory of Combinatorial and Geometric Structures, Moscow Institute of Physics and Technology (National Research Institute), Moscow, Russia

    Alexandr Polyanskii

  3. Institute for Information Transmission Problems RAS, Moscow, Russia

    Alexandr Polyanskii

  4. Caucasus Mathematical Center, Adyghe State University, Maikop, Russia

    Alexandr Polyanskii

Authors
  1. Rom Pinchasi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alexandr Polyanskii
    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: Kenneth Clarkson

Dedicated to the memory of Branko Grünbaum.

Publisher's Note

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

R. Pinchasi: Supported by ISF Grant (Grant No. 409/16). A. Polyanskii: Supported in part by the Leading Scientific Schools of Russia through Grant No. NSh-6760.2018.1.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Pinchasi, R., Polyanskii, A. A One-Page Solution of a Problem of Erdős and Purdy. Discrete Comput Geom 64, 382–385 (2020). https://doi.org/10.1007/s00454-019-00139-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00454-019-00139-1

Keywords

Mathematics Subject Classification

Search

Navigation