Long cycles have the edge-Erd\H{o}s-P\'osa property

We prove that the set of long cycles has the edge-Erd\H{o}s-P'osa property: for every fixed integer $\mathrm{\ell }\ge 3$ and every $k\in \mathbb{N}$, every graph $G$ either contains $k$ edge-disjoint cycles of length at least $\mathrm{\ell }$ (long cycles) or an edge set $X$ of size $O(k^2\log k+\mathrm{\ell }k)$ such that $G-X$ does not contain any long cycle. This answers a question of Birmel'e, Bondy, and Reed (Combinatorica 27 (2007), 135–145).