In the first part of my talk I shall present certain consequences of recent works of Šťovíček and Trlifaj; in particular an improvement of the existing characterization of tilting and cotilting cotorsion pairs. One of the sufficient conditions for a cotorsion pair to be (co-)tilting, completeness, is shown to be redundant.
Is it consistent with ZFC that the cotorsion pair, (A, B), generated by a set of modules is complete? This problem remains still unsolved. However, if we put some extra assumptions on the cotorsion pair (for example 'A is closed under pure submodules', but there are others), then it is possible to prove, assuming V = L, that (A, B) is cogenerated by a set, hence complete. This is (in brief) a content of the second part of my talk.
 J. Šaroch, On a characterization of cotilting cotorsion classes, manuscript.
 J. Šaroch and J. Trlifaj, Cotilting modules and Gödel's axiom of constructibility, to appear in Bull. London Math. Soc.
 J. Šťovíček, All n-cotilting modules are pure-injective, to appear in Proc. Amer. Math. Soc.
 J. Šťovíček and J. Trlifaj, All tilting modules are of countable type, to appear in Bull. London