Tournaments with many Hamilton cycles

The object of interest is the maximum number, $h(n)$, of Hamilton cycles in an $n$-tournament. By considering the expected number of Hamilton cycles in various classes of random tournaments, we obtain new asymptotic lower bounds on $h(n)$. The best result so far is approximately $\mathrm{2.85584...}$ times the expected number $g(n)$ of Hamilton cycles in a random $n$-tournament, and it is conjectured that $h(n)\sim cg(n)$ where $c\approx 2.855958$. The same statements hold for Hamilton paths.