Axiomatizing AECs and applications

For any abstract elementary class (AEC) ${\bf K}$ with $\lambda=LS({\bf K})$, the following holds: 1. $K$ has an axiomatization in $L_{(2^\lambda)^+,\lambda^+}$, allowing game quantification. If ${\bf K}$ has arbitrarily large models, the $\lambda$-amalgamation property and is categorical both in $\lambda$ and $\lambda^+$, then it has an axiomatization in $L_{\lambda^{+},\lambda^{+}}$ with game quantification. These extend Kueker's result which assumes finite character and $\lambda=\aleph_0$. 2. If $K$ is universal and categorical in $\lambda$, then it is axiomatizable in $L_{\lambda^+,\lambda^+}$. 3. Shelah's celebrated presentation theorem asserts that for any AEC ${\bf K}$ there is a first-order theory in an expansion of $L({\bf K})$, and a set $\Gamma$ of $2^\lambda$ many $T$-types such that $K=PC(T,\Gamma,L({\bf K}))$. We provide a better bound on $|\Gamma|$ in terms of $I_2(\lambda,{\bf K})$. 4. We present additional applications which extend, simplify and generalize results of Shelah and Shelah-Vasey. Some of our main results generalize to $\mu$-AECs.