  A celebrated unresolved conjecture of Erd\H{o}s and Hajnal states that for
every undirected graph $H$ there exists $\epsilon(H)>0$ such that every
undirected graph on $n$ vertices that does not contain $H$ as an induced
subgraph contains a clique or stable set of size at least $n^{\epsilon(H)}$.
The conjecture has a directed equivalent version stating that for every
tournament $H$ there exists $\epsilon(H)>0$ such that every $H$-free $n$-vertex
tournament $T$ contains a transitive subtournament of order at least
$n^{\epsilon(H)}$. We say that a tournament is \textit{prime} if it does not
have nontrivial homogeneous sets. So far the conjecture was proved only for
some specific families of prime tournaments (\cite{chorochudber, choromanski2})
and tournaments constructed according to the so-called \textit{substitution
procedure}(\cite{alon}). In particular, recently the conjecture was proved for
all five-vertex tournaments (\cite{chorochudber}), but the question about the
correctness of the conjecture for all six-vertex tournaments remained open. In
this paper we prove that all but at most one six-vertex tournament satisfy the
Erd\H{o}s-Hajnal conjecture. That reduces the six-vertex case to a single
tournament.
