Kelly–Kühn–Osthus conjecture on directed cycle semidegree thresholds

An oriented graph is a directed graph without loops or multiple edges. For an oriented graph $G$, let $\delta^\pm(G)$ be its minimum semidegree, the smaller of its minimum indegree and minimum outdegree. For an integer $\ell\ge4$, let $k$ be the smallest integer greater than $2$ that does not divide $\ell$. Kelly–Kühn–Osthus conjecture. For every $\ell\ge4$ there exists $n_0:=n_0(\ell)$ such that every oriented graph $G$ on $n\ge n_0$ vertices with

$$\delta^\pm(G) \ge \frac{n}{k}+\frac{1}{k}$$

contains a directed cycle of length exactly $\ell$. The conjectured threshold is best possible up to the stated additive term, as shown by balanced blow-ups of a directed $k$-cycle. The conjecture is proved when $k=3$; the paper resolves the remaining cases for cycles of length divisible by $3$ but not equal to $3$.