4-Holes in point sets

We consider a variant of a question of Erd?s 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 2 - ? ( n ) ; and the maximum number of non-convex 4-holes is at least 1 2 n 3 - ? ( n 2 log n ) and at most 1 2 n 3 - ? ( n 2 ) .

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