Bhattacharya, Bhaswar BDas, Sandip2023-05-232023-05-232011-12-012018-07-12https://repository.upenn.edu/handle/20.500.14332/48062Let 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.2011.1173primary 52C1052A10convex hulldiscrete geometryempty convex polygonsErdös-Szekeres theoremRamsey-type resultsApplied MathematicsBusinessMathematicsStatistics and ProbabilityReport