Erdos and Moser posed the problem of determining, for each integer n>0, the greatest integer v(n) such that all tournaments of order n contain the transitive subtournament of order v(n) (denoted TTv(n)). It is known that v(n)=3 for \(\), v(n)=4 for \(\), v(n)=5 for \(\), and \(\) for n>27. Moreover, the uniqueness of the tournaments free of TT4 of orders 6 and 7, and free of TT5 and TT6 of orders 13 and 27, respectively, has been established. Here we prove that the tournaments of orders 12 and 26, free of TT5 and TT6, respectively, are also unique. Then, we see that all tournaments of order 54 contain TT7 (improving the best lower bound known for v(n)). Finally, with the aid of a computer, we obtain the orders cv(r) and gv(s) of the biggest transitive tournaments contained, respectively, in all circulant tournaments of order r≤55 and in each Galois tournament of order s<1000, i.e., in the tournament with set of vertices the Galois field of order s (whenever it exists) and edge directions induced by the quadratic residues. We get better upper bounds of v(n), for n≤991.