A tight Erd\H{o}s-P\'osa function for long cycles

A classic result of Erd\H{o}s and P\'osa says that any graph contains either $k$ vertex-disjoint cycles or can be made acyclic by deleting at most $O(k \log k)$ vertices. Here we generalize this result by showing that for all numbers $k$ and $l$ and for every graph $G$, either $G$ contains $k$ vertex-disjoint cycles of length at least $l$, or there exists a set $X$ of $\mathcal O(kl+k\log k)$ vertices that meets all cycles of length at least $l$ in $G$. As a corollary, the tree-width of any graph $G$ that does not contain $k$ vertex-disjoint cycles of length at least $l$ is of order $\mathcal O(kl+k\log k)$. These results improve on the work of Birmel\'e, Bondy and Reed '07 and Fiorini and Herinckx '14 and are optimal up to constant factors.