Let be a graph with n vertices and edges drawn in the plane. The celebrated Crossing Lemma states that G has at least pairs of crossing edges; or equivalently, there is an edge that crosses other edges. We strengthen the Crossing Lemma for drawings in which any two edges cross in at most points. An ℓ-grid in the drawing of G is a pair of disjoint edge subsets each of size ℓ such that every edge in intersects every edge in . If every pair of edges of G intersect in at most k points, then G contains an ℓ-grid with , where only depends on k. Without any assumption on the number of points in which edges cross, we prove that G contains an ℓ-grid with . If G is dense, that is, , our proof demonstrates that G contains an ℓ-grid with . We show that this bound is best possible up to a constant factor by constructing a drawing of the complete bipartite graph using expander graphs in which the largest ℓ-grid satisfies .
A preliminary version of this paper appeared in the Proceedings of the 15th Symposium on Graph Drawing, Sydney, 2007, in: Lecture Notes in Comput. Sci., vol. 4875, Springer, 2008, pp. 13–24.