Generalized Turán problem for directed cycles

Abstract

For integers $k,\ell \ge 3$, let $\mathrm{ex}(n,\overrightarrow{C_k},\overrightarrow{C_{\ell }})$ denote the maximum number of directed cycles of length $k$ in any oriented graph on $n$ vertices which does not contain a directed cycle of length $\ell$. We establish the order of magnitude of $\mathrm{ex}(n,\overrightarrow{C_k},\overrightarrow{C_{\ell }})$ for every $k$ and $\ell$ and determine its value up to a lower error term when $k\nmid \ell$ and $\ell$ is large enough. Additionally, we calculate the value of $\mathrm{ex}(n,\overrightarrow{C_k},\overrightarrow{C_{\ell }})$ for some other specific pairs $(k,\ell )$ showing that a diverse class of extremal constructions can appear for small values of $\ell$.