On the Minimum Size of a Point Set Containing a 5-Hole and a Disjoint 4-Hole

Let H(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 Ramsey-type result for disjoint empty convex polygons proved by Aichholzer et al. [Graphs and Combinatorics, Vol. 23, 481-507, 2007], Hosono and Urabe [KyotoCGGT, 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.