Kulikov's problem on universal torsion-free abelian groups

Let T be an abelian group and λ an uncountable regular cardinal. The question of whether there is a λ-universal group U among all torsion-free abelian groups G of cardinality less than or equal to A satisfying Ext(G, T) = 0 is considered. U is said to be λ-universal for T if, whenever a torsion-free abelian group G of cardinality at most λ, satisfies Ext(G, T) = 0, there is an embedding of G into U. For large classes of abelian groups T and cardinals λ, it is shown that the answer is consistently no, that is to say, there is a model of ZFC in which, for pairs T and λ, there is no universal group. In particular, for T torsion, this solves a problem by Kulikov.