The $\mathcal{O}(k\log k)$ Erdős–Pósa conjecture for long holes in $C_4$-free graphs

For each $\ell\geq 3$, let $C_\ell$ be the cycle of length $\ell$. A hole is an induced cycle of length at least $4$, and a graph is $C_4$-free if it contains no $C_4$. Let $k\in\mathbb{N}$. Long-hole Erdős–Pósa conjecture. There exists a function $f(k)=\mathcal{O}(k\log k)$ such that for every $C_4$-free graph $G$, either $G$ contains $k$ vertex-disjoint holes of length at least $6$, or there is a set $X$ of at most $f(k)$ vertices such that $G-X$ has no hole of length at least $6$. The paper proves the weaker $\mathcal{O}(k^2\log k)$ bound and identifies this sharper bound as an open problem; its broader final open problem concerns other graph classes with the induced Erdős–Pósa property.