A bipartite strengthening of the Crossing Lemma

Let G = ( V , E ) be a graph with n vertices and m ⩾ 4 n edges drawn in the plane. The celebrated Crossing Lemma states that G has at least Ω ( m 3 / n 2 ) pairs of crossing edges; or equivalently, there is an edge that crosses Ω ( m 2 / n 2 ) other edges. We strengthen the Crossing Lemma for drawings in which any two edges cross in at most O ( 1 ) points. An ℓ-grid in the drawing of G is a pair E 1 , E 2 ⊂ E of disjoint edge subsets each of size ℓ such that every edge in E 1 intersects every edge in E 2 . If every pair of edges of G intersect in at most k points, then G contains an ℓ-grid with ℓ ⩾ c k m 2 / n 2 , where c k > 0 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 ℓ = m 2 / n 2 polylog ( m / n ) . If G is dense, that is, m = Θ ( n 2 ) , our proof demonstrates that G contains an ℓ-grid with ℓ = Ω ( n 2 / log n ) . We show that this bound is best possible up to a constant factor by constructing a drawing of the complete bipartite graph K n , n using expander graphs in which the largest ℓ-grid satisfies ℓ = Θ ( n 2 / log n ) .