On the existence of a point subset with a specified number of interior points

Abstract

An interior point of a finite 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⩾1, let g( k) be the smallest integer such that every set of points in the plane, no three collinear, containing at least g( k) interior points has a subset of points containing exactly k interior points. We prove that g(1)=1, g(2)=4, g(3)⩾8 , and g(k)⩾k+2, k⩾4 . We also give some related results.