Axiomatizing AECs and applications

For any abstract elementary class (AEC) K with λ=LS(K), the following holds: K has an axiomatization in L(2λ)+,λ+, allowing game quantification. If K has arbitrarily large models, the λ-amalgamation property and is categorical both in λ and λ+, then it has an axiomatization in Lλ+,λ+ with game quantification. These extend Kueker's [10] result which assumes finite character and λ=ℵ0. If K is universal and categorical in λ, then it is axiomatizable in Lλ+,λ+. Shelah's celebrated presentation theorem asserts that for any AEC K there is a first-order theory in an expansion of L(K), and a set Γ of 2λ many T-types such that K=PC(T,Γ,L(K)). We provide a better bound on |Γ| in terms of I2(λ,K). We present additional applications which extend, simplify and generalize results of Shelah [13], [15] and Shelah-Vasey [17]. Some of our main results generalize to μ-AECs.