Tue, 18 Oct 2022, 1:25 pm MDT

Let $K$ be an abstract elementary class and $\lambda=\textrm{LS}(K)$. $K$ can be axiomatized by a sentence in $L_{(2^\lambda)^+,\lambda^+}$, allowing game quantification. This is parallel to Shelah-Villaveces result which demands a much higher complexity of junctions but without game quantification. Shelahâ€™s presentation theorem gives $K$ = $\textrm{PC}(T, \Gamma, {L}(K))$ where $T$ is a first-order theory in an expansion of ${L}(K)$ and $\Gamma$ is a set of $T$-types. We provide a better bound of $|\Gamma|$ in terms of $\textrm{I}_2(\lambda,K)$. We also give conditions under which the categoricity in two successive cardinals implies the existence of models in the next cardinal. This improves the result of Shelah.