it has length at least $\ell$. Then for every multigraph $G$ and every nonnegative integer $k$, either $G$ contains $k$ edge-disjoint long cycles or there is a set $F\subseteq E(G)$ of size at most $k \cdot g(k,\ell)$ such that $G-F$ has no long cycle.

Throughout, we abbreviate $g(k):=g(k,\ell)$. Aiming for a contradiction, we assume the theorem is false; let $G$ be a counterexample minimizing $|E(G)|+|V(G)|$ and let $k\geq 1$ be such that $G$ does not contain $k$ edge-disjoint long cycles, nor a hitting set of size at most $k\cdot g(k)$.\
Suppose first that $G$ contains a long cycle $C$ of length at most $g(k)$ and consider the graph $G'=G-E(C)$. Since $G'$ is not a counterexample, it either contains $k-1$ edge-disjoint long cycles, or a set $F'\subseteq E(G')$ of size $|F'| \leq (k-1) \cdot g(k-1)$ such that $G'-F'$ has no long cycles. In the former case we can add $C$ to obtain $k$ edge-disjoint long cycles in $G$; contradiction. In the latter case, $F:=F' \cup E(C)$ is a hitting set for long cycles in $G$, while $F$ has size $|F'|+|E(C)| \leq (k-1) \cdot g(k-1) + g(k) \leq k\cdot g(k)$; contradiction. Therefore we may assume from now on that every long cycle 