Directed graphs without short cycles

For a directed graph $G$ without loops or parallel edges, let $\beta (G)$ denote the size of the smallest feedback arc set, i.e., the smallest subset $X \subset E(G)$suchthat$G \sm X$hasnodirectedcycles\mathrm{.}Let$\gamma(G)$ be the number of unordered pairs of vertices of $G$ which are not adjacent. We prove that every directed graph whose shortest directed cycle has length at least $r\ge 4$ satisfies $\beta (G)\le c\gamma (G)\mathrm{/}{r}^2$, where $c$ is an absolute constant. This is tight up to the constant factor and extends a result of Chudnovsky, Seymour, and Sullivan. This result can be also used to answer a question of Yuster concerning almost given length cycles in digraphs. We show that for any fixed $0<\theta <1\mathrm{/}2$ and sufficiently large $n$, if $G$ is a digraph with $n$ vertices and $\beta(G) \ge \theta n^2$,thenforany$0 \le m \le \theta n-o(n)$ it contains a directed cycle whose length is between $m$ and $m+6{\theta }^{-1\mathrm{/}2}$. Moreover, there is a constant $C$ such that either $G$ contains directed cycles of every length between $C$ and $\theta n-o(n)$ or it is close to a digraph $G^{\mathrm{\prime }}$ with a simple structure: every strong component of $G^{\mathrm{\prime }}$ is periodic. These results are also tight up to the constant factors.