The conjecture that all orientations of odd cycles are avoidable

A digraph $F$ is avoidable if there exists a function $d_F:\mathbb{N}\rightarrow\mathbb{N}$ such that every digraph of minimum out-degree at least $d_F(k)$ contains an $F$-free subdigraph of minimum out-degree at least $k$. An orientation of an odd cycle is a digraph obtained by assigning a direction to every edge of a cycle of odd length.

Odd-cycle avoidability conjecture. All orientations of odd cycles are avoidable.

The paper proves avoidability for all orientations of $C_3$ and $C_5$, while showing that the most direct directed extension of Thomassen's conjecture is false. The conjecture remains open for odd cycles beyond the cases established in the paper.