On topological graphs with at most four crossings per edge

We show that if a graph G with n ≥ 3 vertices can be drawn in the plane such that each of its edges is involved in at most four crossings, then G has at most 6n− 12 edges. This settles a conjecture of Pach, Radoičić, Tardos, and Tóth. As a corollary we also obtain a better bound for the Crossing Lemma which gives a lower bound for the minimum number of crossings in a drawing of a graph.