Weakening the local character
  In [Sh E46], Shelah obtained a non-forking relation for an AEC, (K,\preceq), with LST-number at most \lambda, which is categorical in \lambda and \lambda^+ and has less than 2^{\lambda^+} models of cardinality \lambda^{++}, but at least one. This non-forking relation satisfies the main properties of the non-forking relation on stable first order theories, but only a weak version of the local character.   Here, we improve this non-forking relation such that it satisfies the local character, too. Therefore it satisfies the main properties of the non-forking relation on superstable first order theories.   We conclude that the function \lambda \to I(\lambda,K), which assigns to each cardinal \lambda, the number of models in K of cardinality \lambda, is not arbitrary. 
Adi Jarden and Saharon Shelah
math.LO