Kulikov's problem on universal torsion-free abelian groups revisited

Let T be a torsion abelian group. We consider the class of all torsion-free abelian groups G satisfying Ext(G,T)=0 and search for -universal objects in this class. We show that, for certain T there is no -universal group. However, for uncountable cardinals there is always a -universal group if we assume (V=L). Together with results by the second author this solves completely a problem by Kulikov.