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 ∈n~2 intersecting pairs (disjoint pairs), then there exists two subfamilies F_1, F_2 is contained in F of size δn each, such that every curve in F_1 intersects (is disjoint to) every curve in F_2, where δ depends only on ∈. We apply these results to find pairwise disjoint edges in simple topological graphs.
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