Cycles of a given length in tournaments

We study the asymptotic behavior of the maximum number of directed cycles of a given length in a tournament: let $c(\mathrm{\ell })$ be the limit of the ratio of the maximum number of cycles of length $\mathrm{\ell }$ in an $n$-vertex tournament and the expected number of cycles of length $\mathrm{\ell }$ in the random $n$-vertex tournament, when $n$ tends to infinity. It is well-known that $c(3)=1$ and $c(4)=4\mathrm{/}3$. We show that $c(\mathrm{\ell })=1$ if and only if $\mathrm{\ell }$ is not divisible by four, which settles a conjecture of Bartley and Day. If $\mathrm{\ell }$ is divisible by four, we show that $1+2\cdot\left(2/\pi\right)^{\ell}\le c(\ell)\le 1+\left(2/\pi+o(1)\right)^{\ell}$anddeterminethevalue$c(\ell)$exactlyfor$\ell = 8$. We also give a full description of the asymptotic structure of tournaments with the maximum number of cycles of length $\mathrm{\ell }$ when $\mathrm{\ell }$ is not divisible by four or $\mathrm{\ell }\in \{4,8\}$.