Abstract Assuming the existence of a monster model, tameness, and continuity of nonsplitting in an abstract elementary class (AEC), we extend known superstability results: let $\mu>\operatorname {LS}(\mathbf {K})$ be a regular stability cardinal and let $\chi $ be the local character of $\mu $ -nonsplitting. The following holds: 1. When $\mu $ -nonforking is restricted to $(\mu ,\geq \chi )$ -limit models ordered by universal extensions, it enjoys invariance, monotonicity, uniqueness, existence, extension, and continuity. It also has local character $\chi $ . This generalizes Vasey’s result [37, Corollary 13.16] which assumed $\mu $ -superstability to obtain same properties but with local character $\aleph _0$ . 2. There is $\lambda \in [\mu ,h(\mu ))$ such that if $\mathbf {K}$ is stable in every cardinal between $\mu $ and $\lambda $ , then $\mathbf {K}$ has $\mu $ -symmetry while $\mu $ -nonforking in (1) has symmetry. In this case: (a) $\mathbf {K}$ has the uniqueness of $(\mu ,\geq \chi )$ -limit models: if $M_1,M_2$ are both $(\mu ,\geq \chi )$ -limit over some $M_0\in K_{\mu }$ , then $M_1\cong _{M_0}M_2$ ; (b) any increasing chain of $\mu ^+$ -saturated models of length $\geq \chi $ has a $\mu ^+$ -saturated union. These generalize [31] and remove the symmetry assumption in [10, 38] . Under $(<\mu )$ -tameness, the conclusions of (1), (2)(a)(b) are equivalent to $\mathbf {K}$ having the $\chi $ -local character of $\mu $ -nonsplitting. Grossberg and Vasey [18, 38] gave eventual superstability criteria for tame AECs with a monster model. We remove the high cardinal threshold and reduce the cardinal jump between equivalent superstability criteria. We also add two new superstability criteria to the list: a weaker version of solvability and the boundedness of the U-rank.

We show that ℶ(2LS(K))+ is the lower bound to the Hanf numbers for the length of the order property and for stability in stable abstract elementary classes (AECs). Our examples satisfy the joint embedding property, no maximal model, and (<ℵ0)-tameness but not necessarily the amalgamation property. We also define variations on the order and syntactic order properties insofar as it allows the index set to be linearly ordered rather than well-ordered. Combining with Shelah's stability theorem, we deduce that our examples can have the order property up to any μ<ℶ(2LS(K))+. Boney conjectured that the joint embedding property is needed for two type-counting lemmas. We solved the conjecture by showing it is independent of ZFC.

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.