Numbers of Weights of Convex Quadrilaterals in Weighted Point Sets

Let ℘n denote the family of sets of points in general position in the plane each of which is assigned a different number, called a weight, in {1,2,...,n}. For P∈℘n and a polygon Q with vertices in P, we define the weight of Q as the sum of the weights of its vertices and denote by Wk(P) the set of weights of convex k-gons with vertices in P∈℘n. Let fk(n) = minP∈℘n |Wk(P)|. It is known that n-5 ≤ f₄(n) ≤ 2n-9 for n≥7. In this paper, we show that f₄(n)≥ 4n/3-7.