Polynomial bound for the Directed Grid Theorem

Establish a polynomial upper bound for the function that guarantees a cylindrical grid of order k in every digraph of sufficiently large directed treewidth.

Background

The Directed Grid Theorem guarantees that sufficiently large directed treewidth forces a cylindrical grid, but the best bound currently described in the paper is a power tower of height 22. A polynomial bound would substantially improve the structural understanding of directed treewidth and bring the directed setting closer to the polynomial Grid Theorem for undirected graphs.

References

Obtaining a polynomial bound for the function $f_{\sf DGT}$ remains a challenging open problem.

— Polynomial directed treewidth forces a large bramble with congestion two  (2610.13270 - Campos et al., 9 Oct 2026) in Section 1, paragraph “Brambles in directed graphs”

We ask whether similar ``grid-like'' minors can be obtained in digraphs while maintaining a polynomial dependency on the directed treewidth.

— Polynomial directed treewidth forces a large bramble with congestion two  (2610.13270 - Campos et al., 9 Oct 2026) in Section 1, paragraph “Further research”

Although we were not able to extend this strategy into a construction with polynomial dependence on $k$, we hope that our ideas can be applied to shave a part of the current known bound for cylindrical grids, consisting of a tower of exponentials of height 22, by circumventing webs in the proof.

— Polynomial directed treewidth forces a large bramble with congestion two  (2610.13270 - Campos et al., 9 Oct 2026) in Section 1, paragraph “Further research”