A tight Erdős-Pósa function for planar minors

Let $F$ be a family of graphs. Then for every graph $G$ the maximum number of disjoint subgraphs of $G$, each isomorphic to a member of $F$, is at most the minimum size of a set of vertices that intersects every subgraph of $G$ isomorphic to a member of $F$. We say that $F$ packs if equality holds for every graph $G$. Only very few families pack.

As the next best weakening we say that $F$ has the Erdős-Pósa property if there exists a function $f$ such that for every graph $G$ and integer $k>0$ the graph $G$ has either $k$ disjoint subgraphs each isomorphic to a member of $F$ or a set of at most $f(k)$ vertices that intersects every subgraph of $G$ isomorphic to a member of $F$.

The name is motivated by a classical 1965 result of Erdős and Pósa stating that for every graph $G$ and integer $k>0$ the graph $G$ has either $k$ disjoint cycles or a set of $O(k\log k)$ vertices that intersects every cycle. Thus the family of all cycles has the Erdős-Pósa property with $f(k)=O(k\log k)$. In contrast, the family of odd cycles fails to have the Erdős-Pósa property. For every integer $\ell$, a sufficiently large Escher Wall is a non-bipartite graph that has an embedding in the projective plane such that every face is even and every homotopically non-trivial closed curve intersects the graph at least $\ell$ times. In particular, it contains no set of strictly less than $\ell$ vertices such that each odd cycle contains at least one them, yet it has no two disjoint odd cycles.

By now there is a large body of literature proving that various families $F$ have the Erdős-Pósa property. A very general theorem of Robertson and Seymour says that for every planar graph $H$ the family $F(H)$ of all graphs with a minor isomorphic to $H$ has the Erdős-Pósa property. (When $H$ is non-planar, $F(H)$ does not have the Erdős-Pósa property.) The present paper proves that for every planar graph $H$ the family $F(H)$ has the Erdős-Pósa property with $f(k)=O(k\log k)$, which is asymptotically best possible for every graph $H$ with at least one cycle.