# Many Touchings Force Many Crossings

## TL;DR

This paper establishes a lower bound on the number of crossings among continuous open curves in the plane based on the number of touching pairs, revealing a quadratic relationship under certain conditions.

## Contribution

It provides a tight bound relating touching and crossing pairs in planar curves, advancing understanding of their combinatorial properties.

## Key findings

- Crossings are at least proportional to the square of touches divided by the square of the number of curves.
- The bound is tight apart from constant factors.
- The result applies when the number of touches exceeds ten times the number of curves.

## Abstract

Given $n$ continuous open curves in the plane, we say that a pair is touching if they have only one interior point in common and at this point the first curve does not get from one side of the second curve to its other side. Otherwise, if the two curves intersect, they are said to form a crossing pair. Let $t$ and $c$ denote the number of touching pairs and crossing pairs, respectively. We prove that $c \ge {1\over 10^5}{t^2\over n^2}$, provided that $t\ge 10n$. Apart from the values of the constants, this result is best possible.