Dominant tournament families

For a tournament $H$ with $h$ vertices, its typical density is $h!2^{-\binom{h}{2}}/aut(H)$, i.e. this the expected of in random tournament. A family ${\mathcal F}$ $h$-vertex tournaments {\em dominant} if for all sufficiently large $n$, there exists an $n$-vertex $G$ such that each element larger than by constant factor. Characterizing dominant families challenging already small $h$. Here we characterize several every In particular, prove following large: (i) $H^*$ at least $5\log h$ contain as subgraph dominant. (ii) The whose minimum feedback arc set size most $\frac{1}{2}\binom{h}{2}-h^{3/2}\sqrt{\ln h}$ $h$, construct $6$ (i.e. $50\%$ the) on $5$ vertices and $40\%$ $h=6,7,8,9$. provide explicit construction which conjectured to obtain absolute fraction vertices. Some additional intriguing open problems are presented.