On geometric graphs with no \(k\) pairwise parallel edges (Q1389246)

Two edges of a geometric graph are called parallel if they are the opposite sides of a convex quadrilateral. Main results: if a geometric graph with \(n\) vertices has no pair of parallel edges, then it has at most \(2n-2\) edges. This solves a conjecture of Kupitz. Also, for any fixed \(k\geq 3\), any geometric graph with \(n\) vertices and with no \(k\) pairwise parallel edges contains at most \(O(n)\) edges.