More on Planar Point Subsets with a Specified Number of Interior Points

An interior point of a finite planar point set is a point of the set that is not on the boundary of the convex hull of the set. For any integer $k\ge1$, let $g(k)$ be the smallest integer such that every set $P$ of points in the plane with no three collinear points and with at least $g(k)$ interior points has a subset containing precisely $k$ interior point of $P$. We prove that $g(k)\ge3k$ for $k\ge3$, which improves the known result that $g(k)\ge3k-1$ for $k\ge3$.