Density Theorems for Intersection Graphs of t-Monotone Curves

A curve γ in the plane is t-monotone if its interior has at most t− 1 vertical tangent points. A family of t-monotone curves F is simple if any two members intersect at most once. It is shown that if F is a simple family of n t-monotone curves with at least ǫn intersecting pairs (disjoint pairs), then there exists two subfamilies F1, F2 ⊂ F of size δn each, such that every curve in F1 intersects (is disjoint to) every curve in F2, where δ depends only on ǫ. We apply these results to find pairwise disjoint edges in simple topological graphs with t-monotone edges.