at least $g(k)+1$.

If $G$ has a vertex $v$ of degree at most one, then no cycle visits $v$, so we are done by induction applied to $G-v$. If $G$ has a degree-two vertex $v$ with neighbour $u$, then (since $G$ is triangle-free) we are done by induction applied to the graph obtained by contracting the edge $uv$.

Thus we may assume that $G$ has minimum degree at least $3$ and hence, by Lemma \[lem:mindegreeminor\], $G$ has a minor $G^*$ of minimum degree at least $3\cdot 2^{g(k)/8} = 3k$. Therefore Lemma \[lem:mindegreepacking\] yields $k$ vertex-disjoint cycles in $G^*$. Decontracting $G^*$ back to $G$, we see that these cycles map to $k$ vertex-disjoint cycles in $G$, as desired.

The proof of the edge-variant of Theorem \[thm:vertexcycle\] is verbatim the same, apart from the canonical replacements of vertex sets with edge sets. In particular, at the end of the proof, we still obtain $k$ *vertex*-disjoint cycles in $G$; note that they are edge-disjoint as well.

Edge-Erdős-Pósa for long cycles {#sec:long_cycles}
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In this section we prove the edge-Erdős-Pósa theorem for long cycles.

First we prove two auxiliary lemmas. The first of these lemmas provides a minimum degree condition for finding many vertex-disjoint 