A tight Erdős–Pósa function for long cycles

A classic result of Erdős and Pósa states that any graph either contains k vertex-disjoint cycles or can be made acyclic by deleting at most O(klog⁡k) vertices. Birmelé, Bondy, and Reed (2007) raised the following more general question: given numbers l and k, what is the optimal function f(l,k) such that every graph G either contains k vertex-disjoint cycles of length at least l or contains a set X of f(l,k) vertices that meets all cycles of length at least l? In this paper, we answer that question by proving that f(l,k)=Θ(kl+klog⁡k). 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 O(kl+klog⁡k). This is also optimal up to constant factors and answers another question of Birmelé, Bondy, and Reed (2007).