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 ( n 4 ) ; the minimum number of general 4-holes is at least 5 2 n − Θ(n); and the maximum number of non-convex 4-holes is at least 1 2 n −Θ(n2 log n) and at most 1 2 n −Θ(n2).