  A celebrated unresolved conjecture of Erd\"{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 a 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)} $. Both the directed and the undirected versions of the
conjecture are known to be true for small graphs (tournaments). So far the
conjecture was proved only for some specific families of prime tournaments,
tournaments constructed according to the so$-$called substitution procedure
allowing to build bigger graphs, and for all five$-$vertex tournaments.
Recently the conjecture was proved for all six$-$vertex tournament, with one
exception, but the question about the correctness of the conjecture for all
seven$-$vertex tournaments remained open. In this paper we prove the
correctness of the conjecture for several seven$-$vertex tournaments.
