On the cogeneration of cotorsion pairs
  Let R be a Dedekind domain. Enochs' solution of the Flat Cover Conjecture was extended as follows:   (*) If C is a cotorsion pair generated by a class of cotorsion modules, then C is cogenerated by a set.   We show that (*) is the best result provable in ZFC in case R has a countable spectrum: the Uniformization Principle UP^+ implies that C is not cogenerated by a set whenever C is a cotorsion pair generated by a set which contains a non-cotorsion module. 
Paul C. Eklof and Saharon Shelah and Jan Trlifaj