A note on $\chi$-unbounded classes of geometric graphs

TL;DR

The paper demonstrates that infinitely many classes of geometric intersection graphs, specifically d-CBU graphs for d≥3, are not χ-bounded and are distinct from Burling graphs, resolving an open problem about sources of unbounded chromatic number.

Contribution

It establishes the existence of multiple non-χ-bounded geometric graph classes beyond Burling graphs, answering an open question in the field.

Findings

Existence of infinitely many non-χ-bounded geometric graph classes

d-CBU graphs for d≥3 are not χ-bounded

These classes are incomparable with Burling graphs

Abstract

We show that there exist infinitely many classes of intersection graphs of geometric objects that are not $\chi$-bounded -- namely, $d$-CBU graphs for $d\geq 3$ -- and each is incomparable with the class of Burling graphs. This answers a folklore open problem on whether Burling graphs are the sole source of unbounded chromatic number among geometric intersection classes.