Erdős-Posa property for long directed cycles

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

A directed cycle is a directed closed path with distinct vertices except for its initial and final vertex; its length is its number of arcs. Cycles are disjoint here when they have disjoint vertex sets. For a vertex set $T$, write $D-T$ for vertex deletion from the finite digraph $D$.

Statement recorded in the supplied source.

For every integer $\ell\ge2$ and every integer $n\ge0$, is there an integer $t_n(\ell)\ge0$ such that every digraph $D$ either contains $n$ pairwise vertex-disjoint directed cycles of length at least $\ell$, or has a vertex set $T$ with $|T|\le t_n(\ell)$ for which $D-T$ has no directed cycle of length at least $\ell$? [S2]

Short English statement

Can long directed cycles always be either packed in a prescribed number or hit by a bounded number of vertices?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 3274 (OPG-52200). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Open Problem Garden, Erdős-Posa property for long directed cycles, original node 52200, statement and mathematical discussion. https://www.openproblemgarden.org/op/erdos_posa_property_for_long_directed_cycles.