  Tournaments are orientations of the complete graph, and the directed Ramsey
number $R(k)$ is the minimum number of vertices a tournament must have to be
guaranteed to contain a transitive subtournament of size $k$, which we denote
by $TT_k$. We include a computer-assisted proof of a conjecture by
Sanchez-Flores that all $TT_6$-free tournaments on 24 and 25 vertices are
subtournaments of $ST_{27}$, the unique largest TT_6-free tournament. We also
classify all $TT_6$-free tournaments on 23 vertices. We use these results,
combined with assistance from SAT technology, to obtain the following improved
bounds: $34 \leq R(7) \leq 47$.
