Martin's conjecture for \(\omega\)-stable theories (Q1077407)

Suppose T is a complete \(L_{\omega \omega}\)-theory with less than \(2^{\aleph_ 0}\) countable models. It is shown that the \(L_ 1(T)\)- theory of any countable model is \(\aleph_ 0\)-categorical, where \(L_ 1(T)\) denotes the smallest fragment of \(L_{\omega_ 1\omega}\) containing \(L_{\omega \omega}\) and for each \(n\in \omega\), all complete n-types of T. The proof uses the analysis of the models of such a theory given by \textit{S. Shelah}, \textit{L. Harrington}, and \textit{M. Makkai} [Isr. J. Math. 49, 269-280 (1984; Zbl 0584.03021)].