Density Theorems for Intersection Graphs of $t$-Monotone Curves

Abstract

A curve $\gamma$ 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 $\epsilon n^2$ intersecting pairs (disjoint pairs), then there exists two subfamilies $F_1,F_2\subset F$ of size $\delta n$ each, such that every curve in $F_1$ intersects (is disjoint to) every curve in $F_2$, where $\delta$ depends only on $\epsilon$. We apply these results to find pairwise disjoint edges in simple topological graphs with $t$-monotone edges.