  Let $H_r(n,p)$ denote the maximum number of Hamiltonian cycles in an
$n$-vertex $r$-graph with density $p \in (0,1)$. The expected number of
Hamiltonian cycles in the random $r$-graph model $G_r(n,p)$ is
$E(n,p)=p^n(n-1)!/2$ and in the random graph model $G_r(n,m)$ with
$m=p\binom{n}{r}$ it is, in fact, slightly smaller than $E(n,p)$. For graphs,
$H_2(n,p)$ is proved to be only larger than $E(n,p)$ by a polynomial factor and
it is an open problem whether a quasi-random graph with density $p$ can be
larger than $E(n,p)$ by a polynomial factor. For hypergraphs (i.e. $r \ge 3$)
the situation is drastically different. For all $r \ge 3$ it is proved that
$H_r(n,p)$ is larger than $E(n,p)$ by an {\em exponential} factor and,
moreover, there are quasi-random $r$-graphs with density $p$ whose number of
Hamiltonian cycles is larger than $E(n,p)$ by an exponential factor.
