Linear relation between piercing and independence numbers for planar boxes

Establish whether every finite family of axis-parallel boxes in the plane satisfies a piercing-number bound $\tau(\mathcal{B})=O(\alpha(\mathcal{B}))$, where $\tau(\mathcal{B})$ is the minimum number of points meeting all boxes and $\alpha(\mathcal{B})$ is the independence number of the intersection graph.

Background

For a family of boxes, the piercing number is at least the independence number, but the two parameters need not be equal. The paper notes that Wegner’s proposed two-dimensional bound was recently disproved, while the best general upper bound is only larger than linear by a squared double-logarithmic factor. The unresolved issue is whether a constant-factor relationship still holds in dimension two.

References

On the other hand, it remains open whether $\tau(\mathcal{B})=O(\alpha(\mathcal{B}))$, with the best known general bound $\tau(\mathcal{B})=O(\alpha(\mathcal{B})(\log\log \alpha(\mathcal{B}))2)$ following from a work of Correa, Feuilloley, Perez-Lantero, and Soto .

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