works for a given graph $J$, then essentially the same proof also yields the vertex-EP property. Moreover, if the balls exhibit exponential growth, then this method yields the optimal $O(k \log k)$ bound. If the balls grow slower, e.g. polynomially in the radius, then the method will still yield the edge-EP property, albeit with a larger bounding function.\

Classic Erdős-Pósa {#sec:classicEP}
==================

\[sec:classic\_EP\]

In this section, we present the aforementioned proof of the Erdős-Pósa theorem for cycles, as a warm-up for the proofs presented in section \[sec:long\_cycles\]. It is along the lines of the proof given in [@Raymond2017]. Our main tool is the following powerful lemma of Kühn and Osthus [@KO03], see also Diestel and Rempel [@DR05]. Its short proof is included for completeness.

Given two vertices $u,v$ in a graph $G$, we let $d_G(u,v)$ denote the distance between $u$ and $v$ in $G$.

\[lem:mindegreeminor\] Let $m\geq 1$ and $d\geq 3$ be integers. Then every graph $G$ of girth at least $8m+3$ and minimum degree $d$ contains a minor of minimum degree at least $d(d-1)^m$.

This proof is taken over literally from [@KO03]. For each $x\in V(G)$ and integer $r\geq 0$, let $B_r(x):= \left\{ u \in V(G) \mid d_G(x,u) \leq 