Density theorems for intersection graphs of t-monotone curves

TL;DR

This paper establishes density theorems for intersection graphs of t-monotone curves, showing that large families with many intersections contain large bipartite subfamilies with complete intersection or disjointness.

Contribution

It introduces new density theorems for simple families of t-monotone curves, linking the number of intersections to the existence of large bipartite subfamilies with uniform intersection properties.

Findings

Large families with many intersections contain large bipartite subfamilies with all pairs intersecting.

Similarly, families with many disjoint pairs contain large bipartite subfamilies with all pairs disjoint.

Results apply to finding disjoint edges in simple topological graphs.

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 \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 \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.