Maximal Number of Edges in Geometric Graphs without Convex Polygons

Abstract

A geometric graph G is a graph whose vertex set is a set P _n of n points on the plane in general position, and whose edges are straight line segments (which may cross) joining pairs of vertices of G. We say that G contains a convex r-gon if its vertex and edge sets contain, respectively, the vertices and edges of a convex polygon with r vertices. In this paper we study the following problem: Which is the largest number of edges that a geometric graph with n vertices may have in such a way that it does not contain a convex r-gon? We give sharp bounds for this problem. We also give some bounds for the following problem: Given a point set, how many edges can a geometric graph with vertex set P _n have such that it does not contain a convex r-gon?

A result of independent interest is also proved here, namely: Let P _n be a set of n points in general position. Then there are always three concurrent lines such that each of the six wedges defined by the lines contains exactly $\lfloor \frac{n}{6} \rfloor$ or $\lceil \frac{n}{6} \rceil$ elements of P _n .