Linear chi-bounding function for rectangle intersection graphs

Determine whether the family of intersection graphs of axis-parallel rectangles in the plane admits a linear chi-bounding function f(k)=O(k).

Background

The paper discusses the chi-boundedness of intersection graphs of axis-parallel rectangles in the plane. Although a quadratic bound was known and a later result gives f(k)=O(k log k), it remains unresolved whether the dependence on the clique number can be reduced to linear. This is a general structural graph-theory problem distinct from the paper’s constructions in three-dimensional box-intersection graphs.

References

Demonstrating the surprising complexity of rectangle intersection graphs, it is still open whether there is a linear $\chi$-bounding function $f(k)=O(k)$, with the best known bound $f(k)=O(k\log k)$ being established by Chalermsook and Walczak .

— Geometric triangle-free graphs of large chromatic number  (2610.03517 - Tomon, 2 Oct 2026) in Section 1, subsection “Box-intersection graphs”