Tournaments are orientations of the complete graph. 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 TTk\documentclass[12pt]{minimal}
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\begin{document}$$ TT _k$$\end{document}. We include a computer-assisted proof of a conjecture by Sanchez-Flores in Graphs Combinatorics 14(2), 181–200 (1998), that all TT6\documentclass[12pt]{minimal}
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\begin{document}$$ TT _6$$\end{document}-free tournaments on 24 and 25 vertices are subtournaments of ST27\documentclass[12pt]{minimal}
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\begin{document}$$ ST _{27}$$\end{document}, the unique largest TT6\documentclass[12pt]{minimal}
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\begin{document}$$ TT _6$$\end{document}-free tournament. We also classify all TT6\documentclass[12pt]{minimal}
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\begin{document}$$ TT _6$$\end{document}-free tournaments on 23 vertices. We use these results, combined with assistance from a SAT solver, to obtain the following improved bounds on R(7): 34≤R(7)≤47\documentclass[12pt]{minimal}
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\begin{document}$$34 \le R(7) \le 47$$\end{document}.
