Reconfirmation of Two Results on Disjoint Empty Convex Polygons

For k ≥ 3, let m(k, k + 1) be the smallest integer such that any set of m(k, k + 1) points in the plane, no three collinear, contains two different subsets Q1 and Q2, such that CH(Q1) is an empty convex k-gon, CH(Q2) is an empty convex (k + 1)-gon, and CH(Q1) ∩ CH(Q2) = 0, where CH stands for the convex hull. In this paper, we revisit the case of k = 3 and k = 4, and provide new proofs.