  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)}
$. This 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 size at least
$ n^{\epsilon(H)} $. Recently the conjecture was proved for all six-vertex
tournaments, except $K_{6}$. In this paper we construct two infinite families
of tournaments for which the conjecture is still open for infinitely many
tournaments in these two families $-$ the family of so-called super nebulas and
the family of so-called super triangular galaxies. We prove that for every
super nebula $H_{1}$ and every $\Delta$galaxy $H_{2}$ there exist
$\epsilon(H_{1},H_{2})$ such that every $\lbrace H_{1},H_{2}\rbrace$$-$free
tournament $T$ contains a transitive subtournament of size at least
$\mid$$T$$\mid^{\epsilon(H_{1},H_{2})}$. We also prove that for every central
triangular galaxy $H$ there exist $\epsilon(K_{6},H)$ such that every $\lbrace
K_{6},H\rbrace$$-$free tournament $T$ contains a transitive subtournament of
size at least $\mid$$T$$\mid^{\epsilon(K_{6},H)}$. And we give an extension of
our results.
