# A sharp threshold phenomenon in string graphs

## TL;DR

This paper establishes a threshold phenomenon in string graphs, showing a sharp division between large bipartite disjoint subsets and the existence of small such subsets near the critical density.

## Contribution

It proves a sharp threshold result for the structure of string graphs based on intersection density, revealing a phase transition in their bipartite substructure.

## Key findings

- Below the threshold, large disjoint bipartite subsets exist.
- Above the threshold, only small bipartite subsets can be found.
- The threshold is at approximately one-quarter of the maximum possible intersections.

## Abstract

We prove that for every $\epsilon>0$ there exists $\delta>0$ such that the following holds. Let $\mathcal{C}$ be a collection of $n$ curves in the plane such that there are at most $(\frac{1}{4}-\epsilon)\frac{n^{2}}{2}$ pairs of curves $\{\alpha,\beta\}$ in $\mathcal{C}$ having a nonempty intersection. Then $\mathcal{C}$ contains two disjoint subsets $\mathcal{A}$ and $\mathcal{B}$ such that $|\mathcal{A}|=|\mathcal{B}|\geq \delta n$, and every $\alpha\in \mathcal{A}$ is disjoint from every $\beta\in\mathcal{B}$. On the other hand, for every positive integer $n$ there exists a collection $\mathcal{C}$ of $n$ curves in the plane such that there at most $(\frac{1}{4}+\epsilon)\frac{n^{2}}{2}$ pairs of curves $\{\alpha,\beta\}$ having a nonempty intersection, but if $\mathcal{A},\mathcal{B}\subset \mathcal{C}$ are such that $|\mathcal{A}|=|\mathcal{B}|$ and $\alpha\cap \beta=\emptyset$ for every $(\alpha,\beta)\in \mathcal{A}\times\mathcal{B}$, then $|\mathcal{A}|=|\mathcal{B}|=O(\frac{1}{\epsilon}\log n)$.