A note on trivial nonmultidimensional superstable theories (Q1805404)

Let \(T\) be a nonmultidimensional superstable theory. It was shown by A. Pillay that, if one adds a suitable sequence of parameters to the language, then \(T\) admits \(j\)-constructible (in particular locally atomic) models over arbitrary sets. Pillay raised the question whether these parameters can be removed. Here a positive answer is given when \(T\) is trivial (namely all regular types have trivial pregeometries). Under this assumption, the author also provides some structure results for the models of \(T\), and an upper bound for the spectrum function \((I(\aleph_ \alpha,T) \leq | \alpha + \omega |^{(2^{| T |})})\).