Advertisement

A Note on the Number of General 4-holes in (Perturbed) Grids

  • O. Aichholzer
  • T. Hackl
  • P. Valtr
  • B. VogtenhuberEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9943)

Abstract

Considering a variation of the classical Erdős-Szekeres type problems, we count the number of general 4-holes (not necessarily convex, empty 4-gons) in squared Horton sets of size \(\sqrt{n}\!\times \!\sqrt{n}\). Improving on previous upper and lower bounds we show that this number is \(\varTheta (n^2\log n)\), which constitutes the currently best upper bound on minimizing the number of general 4-holes for any set of n points in the plane.

To obtain the improved bounds, we prove a result of independent interest. We show that \(\sum _{d=1}^n \frac{\varphi (d)}{d^2} = \varTheta (\log n)\), where \(\varphi (d)\) is Euler’s phi-function, the number of positive integers less than d which are relatively prime to d. This arithmetic function is also called Euler’s totient function and plays a role in number theory and cryptography.

Keywords

Grid Point Lattice Line Collinear Point Empty Triangle Prime Segment 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Aichholzer, O., Fabila-Monroy, R., González-Aguilar, H., Hackl, T., Heredia, M.A., Huemer, C., Urrutia, J., Valtr, P., Vogtenhuber, B.: 4-holes in point sets. CGTA 47(6), 644–650 (2014)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Aichholzer, O., Fabila-Monroy, R., González-Aguilar, H., Hackl, T., Heredia, M.A., Huemer, C., Urrutia, J., Valtr, P., Vogtenhuber, B.: On \(k\)-gons and \(k\)-holes in point sets (2014). arXiv:1409.0081
  3. 3.
    Aichholzer, O., Fabila-Monroy, R., González-Aguilar, H., Hackl, T., Heredia, M.A., Huemer, C., Urrutia, J., Valtr, P., Vogtenhuber, B.: On \(k\)-gons and \(k\)-holes in point sets. CGTA 48(7), 528–537 (2015)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Bárány, I., Füredi, Z.: Empty simplices in Euclidean space. Can. Math. Bull. 30, 436–445 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bárány, I., Károlyi, G.: Problems and results around the Erdös-Szekeres convex polygon theorem. In: Akiyama, J., Kano, M., Urabe, M. (eds.) JCDCG 2000. LNCS, vol. 2098, pp. 91–105. Springer, Heidelberg (2001). doi: 10.1007/3-540-47738-1_7 CrossRefGoogle Scholar
  6. 6.
    Bárány, I., Marckert, J.-F., Reitzner, M.: Many empty triangles have a common edge. Discrete Comput. Geom. 50(1), 244–252 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bárány, I., Valtr, P.: Planar point sets with a small number of empty convex polygons. Studia Sci. Math. Hungar. 41(2), 243–266 (2004)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Erdős, P.: Some more problems on elementary geometry. Aust. Math. Soc. Gaz. 5, 52–54 (1978)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Erdős, P., Szekeres, G.: A combinatorial problem in geometry. Compos. Math. 2, 463–470 (1935)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Fabila-Monroy, R., Huemer, C., Mitsche, D.: Empty non-convex and convex four-gons in random point sets. Studia Sci. Math. Hungar. 52(1), 52–64 (2015)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Hardy, G.H., Wright, E.M.: An Introduction to the Theory of Numbers, 5th edn. Oxford University Press, London (1979)zbMATHGoogle Scholar
  12. 12.
    Horton, J.: Sets with no empty convex \(7\)-gons. Can. Math. Bull. 26(4), 482–484 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Valtr, P.: Convex independent sets and 7-holes in restricted planar point sets. Disc. Comp. Geom. 7, 135–152 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Valtr, P.: On the minimum number of empty polygons in planar point sets. Stud. Sci. Math. Hung. 30, 155–163 (1995)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2016

Authors and Affiliations

  • O. Aichholzer
    • 1
  • T. Hackl
    • 1
  • P. Valtr
    • 2
  • B. Vogtenhuber
    • 1
    Email author
  1. 1.Institute for Software TechnologyGraz University of TechnologyGrazAustria
  2. 2.Department of Applied Mathematics and Institute for Computer Science (ITI)Charles UniversityPragueCzech Republic

Personalised recommendations