Weakening the local character

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In [Sh E46], Shelah obtained a non-forking relation for an AEC, (K,), with LST-number at most λ, which is categorical in λand λ+ and has less than 2λ⁺ models of cardinality λ⁺⁺, 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 λ→ I(λ,K), which assigns to each cardinal λ, the number of models in K of cardinality λ, is not arbitrary.