Covering the Convex Quadrilaterals of Point Sets

For a point set P on the plane, a four element subset S ⊂ P is called a 4-hole of P if the convex hull of S is a quadrilateral and contains no point of P in its interior. Let R be a point set on the plane. We say that a point set B covers all the 4-holes of R if any 4-hole of R contains an element of B in its interior. We show that if |R| ≥ 2|B|+5 then B cannot cover all the 4-holes of R. A similar result is shown for a point set R in convex position. We also show a point set R for which any point set B that covers all the 4-holes of R has approximately 2|R| points.