Question 23
Consider $n \ge 6$ points $x_1, x_2, \dots, x_n$ on the plane such that no three of them are collinear. We call a graph with vertices $x_1, x_2, \dots, x_n$ a "road network" if it is connected, each of its edges is a line segment, and no two edges intersect at a point other than their vertices. Prove that there are three road networks $G_1, G_2, G_3$ such that $G_i$ and $G_j$ don't have a common edge for $1 \le i \lt j \le 3$.