Towards a conjecture of Birmel\'e-Bondy-Reed on the Erd\H{o}s-P\'osa property of long cycles

A conjecture of Birmel\'e, Bondy and Reed states that for any integer $\ell\geq 3$, every graph $G$ without two vertex-disjoint cycles of length at least $\ell$ contains a set of at most $\ell$ vertices which meets all cycles of length at least $\ell$. They showed the existence of such a set of at most $2\ell+3$ vertices. This was improved by Meierling, Rautenbach and Sasse to $5\ell/3+29/2$. Here we present a proof showing that at most $3\ell/2+7/2$ vertices suffice.