Triangle-free intersection graphs of line segments with large chromatic number

TL;DR

This paper constructs triangle-free intersection graphs of line segments in the plane with arbitrarily large chromatic number, disproving a longstanding conjecture about the boundedness of chromatic number based on clique number.

Contribution

It provides the first explicit construction of triangle-free segment intersection graphs with unbounded chromatic number, answering Erdos's question negatively.

Findings

Constructed triangle-free segment graphs with arbitrarily large chromatic number

Disproved Scott's conjecture on graphs excluding induced subdivisions

Showed chromatic number is not bounded by clique number in this class

Abstract

In the 1970s, Erdos asked whether the chromatic number of intersection graphs of line segments in the plane is bounded by a function of their clique number. We show the answer is no. Specifically, for each positive integer $k$, we construct a triangle-free family of line segments in the plane with chromatic number greater than $k$. Our construction disproves a conjecture of Scott that graphs excluding induced subdivisions of any fixed graph have chromatic number bounded by a function of their clique number.