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

Arkadiusz Pawlik, Jakub Kozik, Tomasz Krawczyk, Michał Lasoń, Piotr Micek, William T. Trotter, Bartosz Walczak
2014 Journal of combinatorial theory. Series B (Print)  
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.
doi:10.1016/j.jctb.2013.11.001 fatcat:nrsq24zcurbjflw5ojs2j3hcgy