4-Holes in point sets

We consider a variant of a question of Erdős on the number of empty k-gons (k-holes) in a set of n points in the plane, where we allow the k-gons to be non-convex. We show bounds and structural results on maximizing and minimizing the number of general 4-holes, and maximizing the number of non-convex 4-holes. In particular, we show that for n⩾9, the maximum number of general 4-holes is (n4); the minimum number of general 4-holes is at least 52n2−Θ(n); and the maximum number of non-convex 4-holes is at least 12n3−Θ(n2logn) and at most 12n3−Θ(n2).