Density theorems for intersection graphs of t-monotone curves

A curve γin the plane is t-monotone if its interior has at most t-1 vertical tangent points. A family of t-monotone curves F is \emph{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 εn^2 intersecting pairs (disjoint pairs), then there exists two subfamilies F_1,F_2 \subset F of size δn each, such that every curve in F_1 intersects (is disjoint to) every curve in F_2, where δdepends only on ε. We apply these results to find pairwise disjoint edges in simple topological graphs.

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