Skip to main content
Log in

Almost Sharp Bounds on the Number of Discrete Chains in the Plane

  • Original paper
  • Published:
Combinatorica Aims and scope Submit manuscript

Abstract

The following generalisation of the Erdős Unit Distance problem was recently suggested by Palsson, Senger, and Sheffer. For a fixed sequence δ = (δ1, …, δk) of k distances, a (k + 1)-tuple (p1, …, pk+1) of distinct points in ℝd is called a k-chain if ∥pjpj+1∥ = δj for every 1 ≤ jk. What is the maximum number C dk (n) of k-chains in a set of n points in ℝd? Improving the results of Palsson, Senger, and Sheffer, we essentially determine this maximum for all k in the planar case. It is only for k ≡ 1 (mod 3) that the answer depends on the maximum number of unit distances in a set of n points. We also obtain almost sharp results for even k in dimension 3, and propose further generalisations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. P. K. Agarwal, E. Nevo, J. Pach, R. Pinchasi, M. Sharir and S. Smorodinsky: Lenses in arrangements of pseudo-circles and their applications, J. ACM 51 (2004), 139–186.

    Article  MathSciNet  MATH  Google Scholar 

  2. B. Aronov and M. Sharir: Cutting circles into pseudo-segments and improved bounds for incidences, Discrete Comput. Geom. 28 (2002), 475–490.

    Article  MathSciNet  MATH  Google Scholar 

  3. P. Brass: On the maximum number of unit distances among n points in dimension four, Intuitive Geometry 6 (1997), 277–290.

    MathSciNet  MATH  Google Scholar 

  4. P. Erdős: On sets of distances of n points.

  5. P. Erdős: On sets of distances of n points in Euclidean space, Magyar Tud. Akad. Mat. Kutató Int. Közl. 5 (1960), 165–169.

    MathSciNet  MATH  Google Scholar 

  6. S. Gunter, E. A. Palsson, B. Rhodes, S. Senger and A. Sheffer: Bounds on point congurations determined by distances and dot products, 2020, arXiv:2011.15055.

  7. L. Guth and N. H. Katz: On the ErdŐs distinct distances problem in the plane, Ann. of Math. 181 (2015), 155–190.

    Article  MathSciNet  MATH  Google Scholar 

  8. H. Kaplan, J. Matoušek, Z. Safernová and M. Sharir: Unit distances in three dimensions, Combin. Probab. Comput. 21 (2012), 597–610.

    Article  MathSciNet  MATH  Google Scholar 

  9. A. Marcus and G. Tardos: Intersection reverse sequences and geometric applications, J. Combin. Theory Ser. A 113 (2006), 675–691.

    Article  MathSciNet  MATH  Google Scholar 

  10. J. Pach and M. Sharir: Geometric incidences, Towards a theory of geometric graphs, Contemp. Math. vol. 342, Amer. Math. Soc., Providence, RI, 2004, 185–223.

    Google Scholar 

  11. E. A. Palsson, S. Senger and A. Sheffer: On the number of discrete chains, 2019, arXiv:1902.08259.

  12. J. Passant: On ErdŐs chains in the plane, 2020, arXiv:2010.14210.

  13. M. Rudnev: On the number of hinges defined by a point set in ℝ2, 2019, arXiv:1902.05791.

  14. J. Spencer, E. Szemerédi and W. T Trotter: Unit distances in the Euclidean plane, Graph theory and combinatorics, Academic Press, 1984, 294–304.

  15. K. J. Swanepoel: Unit distances and diameters in euclidean spaces, Discrete Comput. Geom. 41 (2009), 1–27.

    Article  MathSciNet  MATH  Google Scholar 

  16. J. Zahl: An improved bound on the number of point-surface incidences in three dimensions, Contrib. Discrete Math. 8 (2013), 100–121.

    MathSciNet  MATH  Google Scholar 

  17. J. Zahl: Breaking the 3/2 barrier for unit distances in three dimensions, Int. Math. Res. Not. 2019 (2019), 6235–6284.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We thank Konrad Swanepoel and Peter Allen for helpful comments on the manuscript. We also thank Dömötör Pálvölgyi for suggesting Proposition 3.2. Research of Andrey Kupavskii is supported by the grant RSF N 21-71-10092, https://rscf.ru/project/21-71-10092/.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Andrey Kupavskii.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Frankl, N., Kupavskii, A. Almost Sharp Bounds on the Number of Discrete Chains in the Plane. Combinatorica 42 (Suppl 1), 1119–1143 (2022). https://doi.org/10.1007/s00493-021-4853-6

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00493-021-4853-6

Mathematics Subject Classification (2010)

Navigation