Skip to main content
SpringerLink
Log in

Almost Empty Monochromatic Quadrilaterals in Planar Point Sets

  • Published:
Mathematical Notes Aims and scope Submit manuscript

Abstract

For positive integers c, s ≥ 1, r ≥ 3, let W r (c, s) be the least integer such that if a set of at least W r (c, s) points in the plane, no three of which are collinear, is colored with c colors, then this set contains a monochromatic r-gon with at most s interior points. As is known, if r = 3, then W r (c, s)=W r (c, s). In this paper, we extend these results to the case r = 4. We prove that W4(2, 1) = 11, W4(3, 2) ≤ 120, and the least integer μ4(c) such that W4(c, μ4(c)) < ∞ is bounded by \(\left\lfloor {\frac{{c - 1}}{2}} \right\rfloor \cdot 2 \leqslant \mu 4\left( c \right) \leqslant 2c - 3\),where c ≥ 2.

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

Similar content being viewed by others

References

  1. O. Aichholzer, F. Aurenhammer, P. Gonzalez-Nava, T. Hackl, C. Huemer, F. Hurtado, H. Krasser, S. Ray, and B. Vogtenhuber, “Matching edges and faces in polygonal partitions,” Comput. Geom. 39 (2), 134–141 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  2. O. Aichholzer, T. Hackl, C. Huemer, F. Hurtado, and B. Vogtenhuber, “Large bichromatic point sets admit empty monochromatic 4-gons,” SIAMJ. DiscreteMath. 23 (4), 2147–2155 (2010).

    MathSciNet  MATH  Google Scholar 

  3. D. Basu, K. Basu, B. B. Bhattacharya, and S. Das, “Almost empty monochromatic triangles in planar point sets,” Discrete Appl.Math. 210, 207–213 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  4. P. Bose and G. Toussaint, “Characterizing and efficiently computing quadrangulations of planar point sets,” Comput. Aided Geom. Design 14 (8), 763–785 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  5. P. Bose and G. Toussaint, “No quadrangulation is extremely odd,” Lecture Notes in Comput. Sci. 1004, 372–381 (1995).

    Article  MathSciNet  Google Scholar 

  6. O. Devillers, F. Hurtado, G. Károlyi, and C. Seara, “Chromatic variants of the Erdős-Szekeres theorem on points in convex position,” Comput. Geom. 26 (3), 193–208 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  7. P. Erdős, “Some more problems on elementary geometry,” Austral.Math. Soc. Gaz. 5 (2), 52–54 (1978).

    MathSciNet  MATH  Google Scholar 

  8. P. Erdős and G. Szekeres, “A combinatorial problem in geometry,” CompositoMath. 2, 464–470 (1935).

    MathSciNet  MATH  Google Scholar 

  9. P. Erdős and G. Szekeres, “On some extremumproblems in elementary geometry,” Ann.Univ. Sci.Budapest, 3–4 (1960).

    Google Scholar 

  10. T. Gerken, “Empty convex hexagons in planar point sets,” Discrete Comput. Geom. 39 (1), 239–272 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  11. C. Grima, C. Hernando, C. Huemer, and F. Hurtado, “On some partitioning problems for two-colored point sets,” in Proc. XIII Encuentros de Geometría Computacional Geometry, Zaragoza, Spain, 2009.

  12. H. Harborth, “Konvexe Fünfecke in ebenen Punktmengen,” Elem.Math. 33, 116–118 (1978).

    MathSciNet  MATH  Google Scholar 

  13. J. D. Horton, “Sets with no empty convex 7-gons,” Can. Math. Bull. 26 (4), 482–484 (1983).

    Article  MathSciNet  MATH  Google Scholar 

  14. V. A. Koshelev, “On the Erdős-Szekeres problem,” DokladyMathematics 76 (1), 603–605 (2007).

    MATH  Google Scholar 

  15. C.M. Nicolas, “The empty hexagon theorem,” Discrete Comput. Geom. 38 (2), 389–397 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  16. H. Nyklová, “Almost empty polygons,” Studia Sci.Math. Hungar. 40(3), 269–286 (2003).

    MathSciNet  MATH  Google Scholar 

  17. R. G. Stanton, “A combinatorial problem on convex regions,” in Proc. Louisiana Conf. Combinatorics, Graph Theory and Computing, Louisiana State Univ., Baton Rouge, La., Congr. Numer. 1, 180–188 (1970).

    Google Scholar 

  18. G. Szekeres and L. Peters, “Computer solution to the 17-point Erdős-Szekeres problem,” Anziam J. 48 (2), 151–164 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  19. G. Tóth and P. Valtr, The Erdős–Szekeres theorem: Upper Bounds and Related Results, Charles Univ., 2004.

    MATH  Google Scholar 

  20. G. Toussaint, “Quadrangulations of planar sets,” in Proc. 4th International Workshop on Algorithms and Data Structures (WADS), Lecture Notes in Computer Science (Springer, New York, 1995), Vol. 955, pp. 218–227.

    MathSciNet  Google Scholar 

  21. P. Valtr, “Convex independent sets and 7-holes in restricted planar point sets,” Discrete Comput.Geom. 7(1), 135–152 (1992).

    Article  MathSciNet  MATH  Google Scholar 

  22. P. Brass, W. Moser, and J. Pach, Research Problems in Discrete Geometry (Springer Science & Business Media, 2006).

    MATH  Google Scholar 

  23. J. Pach and P. K. Agarwal, Combinatorial Geometry (JohnWiley & Sons, 2011).

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics, Tianjin University, Tianjin, China

    L. Liu & Y. Zhang

Authors
  1. L. Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Y. Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to L. Liu.

Additional information

The article was submitted by the authors for the English version of the journal.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liu, L., Zhang, Y. Almost Empty Monochromatic Quadrilaterals in Planar Point Sets. Math Notes 103, 415–429 (2018). https://doi.org/10.1134/S0001434618030082

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0001434618030082

Keywords

Search

Navigation