Non-forking w-good frames

We introduce the notion of a w-good \(\lambda \)-frame which is a weakening of Shelah’s notion of a good \(\lambda \)-frame. Existence of a w-good \(\lambda \)-frame implies existence of a model of size \(\lambda ^{++}\). Tameness and amalgamation imply extension of a w-good \(\lambda \)-frame to larger models. As an application we show:

Theorem 0.1. Suppose\(2^{\lambda }< 2^{\lambda ^{+}} < 2^{\lambda ^{++}}\)and\(2^{\lambda ^{+}} > \lambda ^{++}\). If \(\mathbb {I}(\mathbf {K}, \lambda ) = \mathbb {I}(\mathbf {K}, \lambda ^{+}) = 1 \le \mathbb {I}(\mathbf {K}, \lambda ^{++}) < 2^{\lambda ^{++}}\)and\(\mathbf {K}\)is\((\lambda , \lambda ^+)\)-tame, then\(\mathbf {K}_{\lambda ^{+++}} \ne \emptyset \).

The proof presented clarifies some of the details of the main theorem of Shelah (Isr J Math 126:29–128, 2001) and avoids using the heavy set-theoretic machinery of Shelah (Classification theory for abstract elementary classes, College Publications, Charleston, 2009 [§VII]) by replacing it with tameness.

1. See Definition 3.5 for the definitions of all these notions and Diagram 1 for their comparison in strength.

2. In Sect. 3.1 we present a more detailed discussion regarding the implications in the other direction.

3. See [17, §VII.0.4] for a definition of \(\mu _{unif}\) and some of its properties.

4. Combining further results of Shelah, [21, 7.1] actually gets a good \(\lambda \)-frame and a good \(\lambda ^+\)-frame.

5. Shelah shows in [17, §VI.7.4] that under additional hypothesis weak density implies density.

6. It is clear that a \(good^{-(St,Lc)} \lambda \)-frame is stronger than a w-good \(\lambda \)-frame. It is suspected that symmetry does not follow from the other axioms of a good \(\lambda \)-frame, so we suspect that \(good^{-(St,Lc)} \lambda \)-frames are strictly stronger than w-good \(\lambda \)-frames. The reason we do not mention \(good^{-(St,Lc)} \lambda \)-frames until this point is because they are simply a technical tool developed in [11] to encompass both semi-good frames and almost-good frames.

7. Boney [3] uses tameness for 2-types to extend symmetry, in [5, 6.9] it was established that tameness for 1-types is sufficient. Observe that in this paper the results of [3] are enough since symmetry is not assumed.

8. As mentioned in the introduction, Shelah claims the same conclusion from fewer assumptions (see Fact 1.1 and the two paragraphs above it).

9. In [17, §VI.8.3] Shelah shows, under the hypothesis of Fact 1.1, that \(\mathfrak {s}_{min}\) is an almost good \(\lambda \)-frame. The reason we only show that \(\mathfrak {s}_{min}\) is a w-good \(\lambda \)-frame is because by Sect. 3 this is enough to get a model of size \(\lambda ^{+++}\) and because the known proofs of the other properties use the machinery of [17, §VII] which we avoid.

10. Similarly to Theorem 4.2, this is not the best known result for universal classes, stronger results are obtained in [13].

Authors and Affiliations

1. Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA, USA

   Marcos Mazari-Armida

Corresponding author

Correspondence to Marcos Mazari-Armida.

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