Large planar graph expansions lack the edge-Erdős–Pósa property

Let $H$ be a planar graph, and let an $H$-expansion denote a subdivision of $H$. The edge-Erdős–Pósa property concerns whether, for every graph in the family of $H$-expansions, either there are many pairwise edge-disjoint copies or a bounded edge set meets every copy.

Large-expansion conjecture. There is an integer $c$ such that for every planar graph $H$ of treewidth (or even pathwidth) at least $c$, the family of $H$-expansions does not have the edge-Erdős–Pósa property.

The paper proves this failure for subdivisions of all subcubic trees of sufficiently large pathwidth and expects the analogous result for expansions of sufficiently large grids. The conjecture extends these results to all planar graphs of sufficiently large treewidth, or even pathwidth.