Boney and Grossberg [BG] proved that every nice AEC has an independence relation. We prove that this relation is unique: In any given AEC, there can exist at most one independence relation that satisfies existence, extension, uniqueness and local character. While doing this, we study more generally properties of independence relations for AECs and also prove a canonicity result for Shelah's good frames. The usual tools of first-order logic (like the finite equivalence relation theorem or the type amalgamation theorem in simple theories) are not available in this context. In addition to the loss of the compactness theorem, we have the added difficulty of not being able to assume that types are sets of formulas. We work axiomatically and develop new tools to understand this general framework.Comment: 33 page

We develop a notion of forking for Galois-types in the context of Abstract Elementary Classes (AECs). Under the hypotheses that an AEC K is tame, type-short, and failure of an order-property, we consider Definition 1. Let M 0 ≺ N be models from K and A be a set. We say that theAssuming property (E) (Existence and Extension, see Definition 3.3) we show that this non-forking is a well behaved notion of independence, in particular satisfies symmetry and uniqueness and has a corresponding U-rank. We find conditions for a universal local character, in particular derive superstability-like property from little more than categoricity in a "big cardinal". Finally, we show that under large cardinal axioms the proofs are simpler and the non-forking is more powerful.In [BGKV], it is established that, if this notion is an independence notion, then it is the only one.

Abstract. We show that Shelah's Eventual Categoricity Conjecture for successors follows from the existence of class many strongly compact cardinals. This is the first time the consistency of this conjecture has been proven. We do so by showing that every AEC with LS(K) below a strongly compact cardinal κ is < κ-tame and applying the categoricity transfer of Grossberg and VanDieren [GV06a]. These techniques also apply to measurable and weakly compact cardinals and we prove similar tameness results under those hypotheses. We isolate a dual property to tameness, called type shortness, and show that it follows similarly from large cardinals.

We study when a union of saturated models is saturated in the framework of tame abstract elementary classes (AECs) with amalgamation. We prove: $\mathbf{Theorem}$ If $K$ is a tame AEC with amalgamation satisfying a natural definition of superstability (which follows from categoricity in a high-enough cardinal), then for all high-enough $\lambda$: * The union of an increasing chain of $\lambda$-saturated models is $\lambda$-saturated. * There exists a type-full good $\lambda$-frame with underlying class the saturated models of size $\lambda$. * There exists a unique limit model of size $\lambda$. Our proofs use independence calculus and a generalization of averages to this non first-order context.Comment: 27 page

We combine two notions in AECs, tameness and good λ-frames, and show that they together give a very well-behaved nonforking notion in all cardinalities. This helps to fill a longstanding gap in classification theory of tame AECs and increases the applicability of frames. Along the way, we prove a complete stability transfer theorem and uniqueness of limit models in these AECs.VanDieren [11], which came from the latter's thesis, and says that two different types over a large model must differ over some smaller model. Tameness has been used successfully in categoricity transfers (see Grossberg and VanDieren [12,13] and Lessman [18]) and stability transfer (see Grossberg and VanDieren [11]; Baldwin, Kueker and VanDieren [3]; and Lieberman [19]). Unfortunately, not all AECs are tame as Baldwin and Shelah [5] have constructed an AEC that is not tame from the exact sequences of an almost free, non-Whitehead group which exists in ZFC at ℵ 1 and consistently exists in all cardinals. On the other hand, the author has shown in [6] that tameness follows for all AECs from large cardinals. Theorem 1.2 ([6]).• If K is an AEC with LS(K) < κ and κ is strongly compact, then K is < κ tame.• Suppose there is a proper class of strongly compact cardinals. Then every AEC is tame.1450007-2 J. Math. Log. 2014.14. Downloaded from www.worldscientific.com by FLINDERS UNIVERSITY LIBRARY on 02/05/15. For personal use only. Tameness and extending framesThese assumptions commonly appear in addition to tameness: amalgamation is used to make types well behaved and joint embedding then follows from λ-joint embedding. However, these global assumptions are in contrast to the project of frames, which aims to inductively build up a structure theory, cardinal by cardinal, and derive these properties along the way with the aid of weak diamond. On the other hand, the existence of frames in the most general setting (see [26, II, Sec. 3]) uses categoricity in two successive cardinals (and more). If we add no maximal models to this hypothesis, this is already enough to apply the full categoricity transfer of [13].On the other hand, the combination of these hypotheses gives much more than just the sum of their parts. Despite the categoricity transfer results under a tameness hypothesis, there is no robust independence notion for these classes. The closest approximation is likely Boney and Grossberg [7], where an independence notion of "< κ satisfiability" is developed. Although this notion is well-behaved, additional methods beyond tameness are needed. Using the method in this paper, we have an independence notion for tame and categorical AECs under some very mild cardinal arithmetic assumptions; see Theorem 8.3. Looking at good λ-frames, the method for building larger frames is a complicated process that changes the Abstract Elementary Class and drops many of the models; see [26], especially II, Sec. 9.1. Although this is fine for the end goal, a process that deals with the whole class would likely have more applications. We provide such a process for tam...