Numbers of Weights of Convex Quadrilaterals in Weighted Point Sets

Sakai, Toshinori (Tokai University)
Matsumoto, Satoshi

Indonesian Journal of Combinatorics
Vol 8, No 1 (2024)

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.