An Erdős-Pósa theorem for cycles and faces of distinct lengths

We show that for every $k \in \mathbb{N}$, every graph $G$ contains $k$ vertex-disjoint cycles of different lengths, or there exists a set $X \subseteq V(G)$ with $|X| \in \mathcal{O}(k^6\mathsf{polylog}(k))$ such that $G-X$ has at most $k-1$ cycle lengths. We also prove analogous results for facial lengths of embedded graphs. Let $G$ be a graph with a closed 2-cell embedding $\psi$ on a surface $\Sigma$ of Euler genus $g$, let $c$ be a colouring of the faces $\mathcal{F}(\psi)$ of $\psi$, and let $R(G,\psi)$ be the radial graph of $(G, \psi)$. Then there exist $k$ faces $F_1, \ldots , F_k \in \mathcal{F}(\psi)$ that are given pairwise distinct colours by $c$ and are pairwise at distance at least $d$ in $\psi$, or there exists a set $X \subseteq V(G)$ of order at most $\mathcal{O}(k^2dg)$ such that $|\{ c(F) \mid F \in \mathcal{F}(\psi) \text{ and } V(F) \cap \bigcup_{x \in X} N^d_{R(G,\psi)}(x) = \emptyset \}| \leq k(k+2)$. Finally, using a result from additive combinatorics, we show that there are subdivided ladders with only a small number of cycle lengths. This suggests that it may be difficult to improve our bounds.