'Erdős-Pósa from ball packing'
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Introduction
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By a classic theorem of Erdős and Pósa [@Erdos1965independent], every graph either contains $k$ vertex-disjoint cycles or a set of $O(k \log k)$ vertices that meets every cycle. This result has sparked a vast body of generalizations and variations regarding the so-called Erdős-Pósa property. A class of graphs $\mathcal{H}$ is said to have the *vertex-EP property* (resp. *edge-EP property*) if there exists a *bounding function* $f(k): \mathbb{N}\rightarrow \mathbb{R}_{\geq 0}$ such that for all $k\in \mathbb{N}$, every graph $G$ either contains $k$ vertex-disjoint (resp. edge-disjoint) subgraphs in $\mathcal{H}$, or a set $F$ of at most $f(k)$ vertices (resp. edges) such that $G-F$ has no subgraph in $\mathcal{H}$.

The result of Erdős and Pósa implies that the class of cycles in fact has both the vertex and the edge-EP property (cp. [@Diestel]). Despite this, most research since then has focused on the vertex-EP property. For instance, by a seminal result of Robertson and Seymour [@robertson1986graph], the class of $J$-expansions has the vertex-EP property if and only if $J$ is a planar graph. Here, given a graph $J$, a *$J$-expansion* is a graph that contains $J$ as a minor. Recently [@CHJR19] it has been established that 