A Note on the Value in the Disjoint Convex Partition Problem

Let $P$ be a planar point set with no three points collinear; $k$ points of $P$ form a $k$-hole of $P$ if these $k$ points are the vertices of a convex polygon whose interior contains no points of $P$. In this article, we prove that any planar point set containing at least 13 points with no three points collinear contains pairwise disjoint 3-, 4-, and 5-holes if there exists a separating line $SL_{4}$.