Date of this Version
Studia Scientiarum Mathematicarum Hungarica
Let H(k; l), k ≤ l denote the smallest integer such that any set of H(k; l) points in the plane, no three on a line, contains an empty convex k-gon and an empty convex l-gon, which are disjoint, that is, their convex hulls do not intersect. Hosono and Urabe [JCDCG, LNCS 3742, 117–122, 2004] proved that 12 ≤ H(4, 5) ≤ 14. Very recently, using a Ramseytype result for disjoint empty convex polygons proved by Aichholzer et al. [Graphs and Combinatorics, Vol. 23, 481–507, 2007], Hosono and Urabe [Kyoto CGGT, LNCS 4535, 90–100, 2008] improve the upper bound to 13. In this paper, with the help of the same Ramsey-type result, we prove that H(4; 5) = 12.
Originally published in Studia Scientiarum Mathematicarum Hungarica © 2011 Akadémiai Kiadó
This is a pre-publication version. The final version is available at http://dx.doi.org/10.1556/SScMath.2011.1173
primary 52C10, 52A10, convex hull, discrete geometry, empty convex polygons, Erdös-Szekeres theorem, Ramsey-type results
Bhattacharya, B. B., & Das, S. (2011). On the Minimum Size of a Point Set Containing a 5-Hole and a Disjoint 4-Hole. Studia Scientiarum Mathematicarum Hungarica, 48 (4), 445-457. http://dx.doi.org/10.1556/SScMath.2011.1173
Date Posted: 25 October 2018
This document has been peer reviewed.