The planar-minor Erdős–Pósa conjecture with a tight logarithmic bound

Let $H$ be a fixed planar graph. For a graph $G$, let $u_H(G)$ be the maximum number of pairwise vertex-disjoint $H$-models in $G$, and let $\tau_H(G)$ be the minimum size of a vertex set $X$ such that $G-X$ has no $H$-model. A bounding function is a function $f$ satisfying

$$\tau_H(G)\leq f(\nu_H(G))$$

for every graph $G$.

The planar-minor Erdős–Pósa conjecture. There is a constant depending on $H$ such that

$$\tau_H(G)=O(\nu_H(G)\log \nu_H(G)).$$

Equivalently, the Erdős–Pósa property for $H$-models has a $O(k\log k)$ bounding function. This would be tight up to the constant factor whenever $H$ has a cycle, matching the known $\Omega(k\log k)$ lower bound; for forests, the optimal order is already $O(k)$.