Beyond the Richter-Thomassen Conjecture

TL;DR

This paper proves a new crossing lemma for closed curves in the plane, establishing a lower bound on crossing points relative to touching points, and confirms a long-standing conjecture that the total intersections grow quadratically with the number of curves.

Contribution

It introduces a Crossing Lemma for closed curves and proves the Richter-Thomassen conjecture on the quadratic growth of total intersection points.

Findings

Number of crossing points exceeds touching points by at least a logarithmic factor.

Total intersection points among n curves are at least nearly n^2.

Confirms the quadratic lower bound on total intersections for pairwise intersecting closed curves.

Abstract

If two closed Jordan curves in the plane have precisely one point in common, then it is called a {\em touching point}. All other intersection points are called {\em crossing points}. The main result of this paper is a Crossing Lemma for closed curves: In any family of $n$ pairwise intersecting simple closed curves in the plane, no three of which pass through the same point, the number of crossing points exceeds the number of touching points by a factor of at least $\Omega((\log\log n)^{1/8})$. As a corollary, we prove the following long-standing conjecture of Richter and Thomassen: The total number of intersection points between any $n$ pairwise intersecting simple closed curves in the plane, no three of which pass through the same point, is at least $(1-o(1))n^2$.