For sufficiently large $n$, it is known that most [tournaments](https://en.wikipedia.org/wiki/Tournament_(graph_theory)) of size $n$ contains no nontrivial automorphism, though I forgot the reference.


For sufficiently large $n$, does there always exist a tournament of size $n$ such that 


* any subtournament of size at least $\sqrt n$ (hence, including the tournament itself) contains no nontrivial automorphism
* there exists no node with outdegree $n-1$



Tournament whose large subtournaments contain no automorphism
In the edited version, how about just taking a transitive tournament and flipping the edge between the vertices of outdegree 0 and $(n-1)$? I think you can replace the $\sqrt{n}$ in the question by 4. The proof divides into cases based on whether the flipped edge is in the subgraph. If not, the subgraph is transitive. If yes, you can uniquely identify the flipped edge as being involved in all 3-cycles, so it can't be moved by an automorphism. Then it's easy to show nothing else can be moved either.


