Intersection patterns of curves
Jacob Fox, János Pach, Csaba D. Tóth · Journal of the London Mathematical Society · 2011
We prove that for every k ∈ ℕ there is a constant ck > 0 with the following property. Every set of n > 1 continuous curves in the plane, any pair of which intersect in at most k points, has two disjoint subsets A and B, each of size at least ckn, such that either every curve in A intersects all curves in B, or no curve in A intersects any curve in B. This statement does not remain true if we drop the condition on the number of intersection points per pair.