Abstract
Let n(k, l,m), k ≤ l ≤ m, be the smallest integer such that any finite planar point set which has at least n(k, l,m) points in general position, contains an empty convex k-hole, an empty convex l-hole and an empty convex m-hole, in which the three holes are pairwise disjoint. In this article, we prove that n(4, 4, 5) ≤ 16.
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References
P. Erdős and G. Szekeres, “A combinatorial problem in geometry,” Compositio Math. 2, 463–470 (1935).
P. Erdős and G. Szekeres, “On some extremum problem in elementary geometry,” Ann.Univ. Sci. Budapest. Eötvös Sec. Math. 3–4, 53–62 (1960).
J. D. Kalbfleisch, J. G. Kalbfleisch, and R. G. Stanton, “A combinatorial problem on convex regions,” in Proc. Louisiana Conf. on Combinatorics Graph Theory and Computering. Vol. 1. Congressus Numerantium (Louisiana State Univ., Baton Rouge, LA., Congr. Number., 1970), pp. 180–188.
G. Szekeres and L. Peters, “Computer solution to the 17-point Erdős-Szekeres problem,” ANZIAM J. 48(2), 151–164 (2006).
H. Harborth, “Konvexe Fünfecke in ebenen Punktmengen,” Elem. Math. 33(5), 116–118 (1978).
J. D. Horton, “Sets with no empty convex 7-gons,” Canad. Math. Bull. 26(4), 482–484 (1983).
T. Gerken, “Empty convex hexagons in planar point sets,” Discrete Comput. Geom. 39(1-3), 239–272 (2008).
C. M. Nicolás, “The empty hexagon theorem,” Discrete Comput. Geom. 38(2), 389–397 (2007).
M. Urabe, “On a partition into convex polygons,” Discrete Appl. Math. 64(2), 179–191 (1996).
K. Hosono and M. Urabe, “A minimal planar point set with specified disjoint empty convex subsets,” in Computational Geometry and Graph Theory, Lecture Notes in Comput. Sci. (Springer-Verlag, Berlin, 2008), Vol. 4535, pp. 90–100.
K. Hosono and M. Urabe, “On the number of disjoint convex quadrilaterals for a planar point set,” Comput. Geom. 20(3), 97–104 (2001).
K. Hosono and M. Urabe, “On theminimum size of a point set containing two non-intersecting empty convex polygons,” in Discrete and Computational Geometry, Lecture Notes in Comput. Sci. (Springer-Verlag, Berlin, 2005), Vol. 3742, pp. 117–122.
L. Wu and R. Ding, “Reconfirmation of two results on disjoint empty convex polygons,” in Discrete Geometry, Combinatorics and Graph Theory, Lecture Notes in Comput. Sci. (Springer-Verlag, Berlin, 2007), Vol. 4381, pp. 216–220.
B. B. Bhattacharya and S. Das, “On the minimum size of a point set containing a 5-hole and a disjoint 4-hole,” Studia Sci. Math. Hungar. 48(4), 445–457 (2011).
B. B. Bhattacharya and S. Das, “Disjoint empty convex pentagons in planar point sets,” Period. Math. Hungar 66(1), 73–86 (2013).
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Published in Russian in Matematicheskie Zametki, 2014, Vol. 96, No. 2, pp. 285–293.
The text was submitted by the authors in English.
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You, X., Wei, X. A note on the upper bound for disjoint convex partitions. Math Notes 96, 268–274 (2014). https://doi.org/10.1134/S0001434614070281
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DOI: https://doi.org/10.1134/S0001434614070281