Proof. Since $\mu $ has countable cofinality, we can pick $\langle \mu _i : i < \omega \rangle $ increasing cofinal in $\mu $ such that $\mu _0 \ge \text {LS} (\mathbf {K})$ and $\mu _i$ is a categoricity cardinal for each $i < \omega $ . Now we build an increasing chain $\langle M_i : i < \omega \rangle $ such that $M_i \in {\mathbf {K}}_{\mu _i}$ for all $i < \omega $ . Then the union of the chain will be in ${\mathbf {K}}_\mu $ . For $i = 0$ , take any $M_0 \in {\mathbf {K}}_{\mu _0}$ . For $i = j + 1$ , given $M_j$ , first pick any $N_i \in {\mathbf {K}}_{\mu _i}$ . Pick $N_i^0 \le _{\mathbf {K}} N_i$ with $N_i^0 \in {\mathbf {K}}_{\mu _j}$ . By categoricity, there is an isomorphism $f: N_i^0 \cong M_j$ . With some renaming, we can extend this isomorphism to $g: N_i \cong M_i$ , for some $M_i$ with $M_j \le _{\mathbf {K}} M_i$ . Then $M_i \in {\mathbf {K}}_{\mu _i}$ , as desired. ⊣

Proof. Pick an existence triple $(a, M, N)$ in ${\mathbf {K}}_{\text {LS} (\mathbf {K})}$ . Now build $\langle M_i : i \le \mu \rangle $ , $\langle N_i : i \le \mu \rangle $ both increasing continuous such that for every $i < \mu $ :

1. (1) $M_0 = M$ , $N_0 = N$ .

2. (2) $M_i, N_i \in {\mathbf {K}}_{|i| + \text {LS} (\mathbf {K})}$ .

3. (3) $M_i \le _{\mathbf {K}} N_i$ .

4. (4) $a \in N_i \backslash M_i$ .

5. (5) $M_{i} <_{\mathbf {K}} M_{i + 1}$ .

This is possible using the weak extension property at successor stages and taking unions at limit stages. In the end, $M_\mu <_{\mathbf {K}} N_\mu $ , since $a \in N_\mu \backslash M_\mu $ . Thus $M_\mu $ is not maximal. Since $\langle M_i : i \le \mu \rangle $ was strictly increasing, $M_\mu \in {\mathbf {K}}_\mu $ .⊣

Proof. Let $\lambda := \text {LS} (\mathbf {K})$ . Let $(a, M, N)$ be an existence triple in ${\mathbf {K}}_{\lambda }$ . We want to find a strict extension of $(a, M, N)$ . We build $\langle M_i : i \le \lambda ^+ \rangle $ increasing continuous and $\langle a_i : i < \lambda ^+ \rangle $ such that for all $i < \lambda ^+$ :

1. (1) $(a_i, M_i, M_{i + 1})$ is an existence triple in ${\mathbf {K}}_{\lambda }$ .

2. (2) There exists an isomorphism $f_i : N\hspace{-1.5pt} \cong\hspace{-1.5pt} M_{i + 1}$ so that $f_i[M]\hspace{-1.5pt} = \hspace{-1.5pt}M_i$ and $f_i (a)\hspace{-1.5pt} =\hspace{-1.5pt} a_i$ .

This is possible by categoricity. This is enough: since ${\mathbf {K}}_{\lambda ^+}$ has no maximal models, there exists $M' \in {\mathbf {K}}_{\lambda ^+}$ with $M_{\lambda ^+} <_{\mathbf {K}} M'$ . Let $\langle M_i^{\prime } : i \le \lambda ^+ \rangle $ be an increasing continuous resolution of $M'$ . Let $C \subseteq \lambda ^+$ be a club such that $i \in C$ implies $M_i \le _{\mathbf {K}} M_i^{\prime }$ and moreover $M_{\lambda ^+} \cap M_i^{\prime } = M_i$ . Pick $b \in M' \backslash M_{\lambda ^+}$ and pick $i \in C$ so that $b \in M_i^{\prime }$ . Then $M_i <_{\mathbf {K}} M_i^{\prime }$ and one can pick $j < \lambda ^+$ so that $M_{i + 1} <_{\mathbf {K}} M_j^{\prime }$ . Then $(a_i, M_i^{\prime }, M_j^{\prime })$ is a strict extension of $(a_i, M_i, M_{i + 1})$ . Taking an isomorphic copy, we obtain a strict extension of the original triple $(a, M, N)$ .⊣

Proof. Note that for every $\lambda \in [\text {LS} (\mathbf {K}), \mu )$ , both ${\mathbf {K}}_{\lambda }$ and ${\mathbf {K}}_{\lambda ^+}$ are not empty and have no maximal models (we are using that $\mu $ is limit to deduce this for ${\mathbf {K}}_{\lambda ^+}$ ). In particular, there is an existence triple in ${\mathbf {K}}_{\text {LS} (\mathbf {K})}$ . Moreover by Lemma 3.5 (applied to ${\mathbf {K}}_{\ge \lambda }$ ), for any $\lambda \in [\text {LS} (\mathbf {K}), \mu )$ , ${\mathbf {K}}_\lambda $ has the weak extension property. By Lemma 3.4, we get the result. ⊣

Proof. By Theorem 3.6, not every model in ${\mathbf {K}}_\mu $ is maximal. By categoricity in $\mu $ , this implies that ${\mathbf {K}}_\mu $ has no maximal models, hence that ${\mathbf {K}}_{\mu ^+} \neq \emptyset $ . ⊣

Fact 4.2. Let $\mathbf {K}$ be an AEC. For any $M \in \mathbf {K}$ and any regular cardinal $\theta> \text {LS} (\mathbf {K})$ , there exists $N \in \mathbf {K}$ such that $M \le _{\mathbf {K}} N$ , N is locally $\theta $ -model-homogeneous, and $\|N\| \le \|M\|^{<\theta }$ .

Proof. Let $M \in {\mathbf {K}}_\mu $ . Fix $\theta < \mu $ , $N \in {\mathbf {K}}_\mu $ with $M \le _{\mathbf {K}} N$ , and $M_0, N_0 \in {\mathbf {K}}_{\le \theta }$ with $M_0 \le _{\mathbf {K}} M$ , $M_0 \le _{\mathbf {K}} N_0 \le _{\mathbf {K}} N$ . We have to see that $N_0$ embeds inside M over $M_0$ . Without loss of generality, $\theta \ge \text {LS} (\mathbf {K})$ . By assumption, we can pick a categoricity cardinal $\lambda < \mu $ such that $\lambda = \lambda ^\theta $ . Let $M_0^{\prime } \le _{\mathbf {K}} M$ be such that $M_0 \le _{\mathbf {K}} M_0^{\prime }$ and $M_0^{\prime } \in {\mathbf {K}}_{\lambda }$ . By categoricity in $\lambda $ and Fact 4.2, $M_0^{\prime }$ is locally $\theta ^+$ -model-homogeneous. In particular, $N_0$ embeds into $M_0^{\prime }$ over $M_0$ . Thus $N_0$ embeds into M over $M_0$ , as desired. ⊣

Proof. Suppose not. Then for each $i < \delta $ , there exists $M_i^1, M_i^2 \in {\mathbf {K}}_{<\theta }$ and $f_i : M_i^1 \cong _{A_i} M_i^2$ so that $A_i \subseteq M_i^\ell \le _{\mathbf {K}} M$ , $\ell = 1,2$ and $f_i (p \upharpoonright M_i^1) \neq p \upharpoonright M_i^2$ .

By $(<\chi )$ -tameness, we can find $A_i^1 \subseteq M_i^1$ of cardinality strictly less than $\chi $ so that $f_i (p \upharpoonright A_i^1) \neq p \upharpoonright A_i^2$ (we have set $A_i^2 := f_i[A_i^1])$ .

Let $A := \bigcup _{i < \delta } (A_i^1 \cup A_i^2)$ . Recall that p is realized in an extension of M. Moreover, if $\delta < \chi $ then $|A| < \chi $ and if $\delta \ge \chi $ then $|A| \le \delta $ . In either case, M is locally $|A|^+$ -saturated, so $p \upharpoonright A$ is realized in M, say by a. Since $\delta $ is a limit ordinal, there exists $i < \delta $ such that $a \in A_i$ . Now, fix an extension $g_i : M \cong M'$ of $f_i$ (so $M_i^2 \le _{\mathbf {K}} M'$ ). We have:

$$\begin{align*}f_i (p \upharpoonright A_i^1) = g_i (\mathbf{tp} (a / A_i^1; M)) = \mathbf{tp} (g_i (a) / A_i^2; M') = \mathbf{tp} (a / A_i^2; M'). \end{align*}$$

Where we have used that $g_i$ fixes $A_i$ , so $g_i (a) = a$ . On the other hand, since $A_i^2 \subseteq A$ ,

$$\begin{align*}p \upharpoonright A_i^2 = \mathbf{tp} (a / A_i^2; M). \end{align*}$$

Finally, observe that $M_i^2 \le _{\mathbf {K}} M$ and $M_i^2 \le _{\mathbf {K}} M'$ by construction. Therefore since $a \in A_i \subseteq M_i^2$ , $\mathbf {tp} (a / A_i^2; M') = \mathbf {tp} (a / A_i^2; M_i^2) = \mathbf {tp} (a / A_i^2; M)$ . We have shown that $f_i (p \upharpoonright A_i^1) = p \upharpoonright A_i^2$ , contradicting the definition of $A_i^1, A_i^2$ , and $f_i$ .⊣

Proof. Fix $A \subseteq N$ with $|A| \le \|M_0\|$ . Pick $N_0 \le _{\mathbf {K}} N$ with $A \subseteq N_0$ and $\|N_0\| \le \|M_0\|$ . Since $M_1$ is locally universal over $M_0$ , there exists $f: N_0 \xrightarrow [M_0]{} M_1$ . By the definition of nonsplitting (where $M_1, M_2$ there stand for $N_0$ , $f[N_0]$ here), we must have that $f (p \upharpoonright N_0) = p \upharpoonright f[N_0]$ and $f (q \upharpoonright N_0) = q \upharpoonright f[N_0]$ . Since $p \upharpoonright M_1 = q \upharpoonright M_1$ , $f \upharpoonright f[N_0] = q \upharpoonright f[N_0]$ . Therefore $f (p \upharpoonright N_0) = f (q \upharpoonright N_0)$ , hence $p \upharpoonright N_0 = q \upharpoonright N_0$ . ⊣

Proof. Use the hypothesis to fix $f: N_0 \xrightarrow [M_1]{} M$ . Let $q_0 := f^{-1} (p \upharpoonright f[N_0])$ . By invariance, $q_0$ does not split over $M_0$ and $q_0 \upharpoonright M_1 = p \upharpoonright M_1$ . To see the moreover part, assume that $q_0$ is algebraic. Then $p \upharpoonright f[N_0]$ is algebraic, hence p is algebraic. ⊣

Proof. Let $N \in {\mathbf {K}}_{\mu }$ and let $M \in {\mathbf {K}}_{[\text {LS} (\mathbf {K}), \mu )}$ with $M \le _{\mathbf {K}} N$ . Pick an increasing sequence $\langle \mu _i : i < \omega \rangle $ cofinal in $\mu $ with $\|M\| \le \mu _0$ and so that for any $i < \omega $ , $\mu _{i + 1}^{\mu _i} = \mu _{i + 1}$ and $\mathbf {K}$ is categorical in $\mu _i$ . This is possible by assumption. By increasing M if needed, we can assume without loss of generality that $M \in {\mathbf {K}}_{\mu _0}$ . Now pick any increasing sequence $\langle N_i : i < \omega \rangle $ such that $N_0 = M$ , $\bigcup _{i < \omega } N_i = N$ , and $N_i \in {\mathbf {K}}_{\mu _i}$ for all $i < \omega $ . We claim this is a nice filtration of N over M.

As in the proof of Lemma 4.3, for any $i < \omega $ , $N_{i + 1}$ is locally universal over $N_i$ . Also, $\bar {N} = \langle N_i : i < \omega \rangle $ is trivially local since there are no limit ordinals below $\omega $ . Furthermore, $\bar {N}$ is coherent because by assumption ${\mathbf {K}}_{<\mu }$ has coherent sequences. Finally, pick $p \in \mathbf {S} (N)$ . Note that by Lemma 4.3, N is locally model-homogeneous, hence locally saturated. Thus Lemma 4.6 (where $\delta , \chi $ there stands for $\omega , \text {LS} (\mathbf {K})^+$ here) implies there exists $i < \omega $ so that p does not split over $N_i$ . ⊣

Proof. Fix $\bar {N} = \langle N_i : i < \delta \rangle $ a nice filtration of N over $M_1$ . For $i < \delta $ , let $q_i \in \mathbf {S} (N_i)$ be as given by weak extension: it extends $p \upharpoonright M_1$ and does not split over $M_0$ . Moreover it is nonalgebraic if p is nonalgebraic. By weak uniqueness (and tameness), $q_j \upharpoonright N_i = q_i$ for $i < j < \delta $ . By Fact 2.7, there exists $q \in \mathbf {S} (N)$ such that $q \upharpoonright N_i = q_i$ for all $i < \delta $ . Note that q cannot be algebraic if all the $q_i$ ’s are nonalgebraic. By the properties of $\bar {N}$ , there exists $i < \delta $ so that q does not split over $N_i$ .

Claim: q does not split over $M_0$ .

Proof of Claim: By Remark 4.5, it suffices to see that $q \upharpoonright N_0^{\prime }$ does not split over $M_0$ for any $N_0^{\prime } \in {\mathbf {K}}_{<\|N\|}$ with $N_{i + 1} \le _{\mathbf {K}} N_0^{\prime } \le _{\mathbf {K}} N$ . So let $q_0^{\prime } \in \mathbf {S} (N_0^{\prime })$ be as given by weak extension (extending $p \upharpoonright M_1$ and not splitting over $M_0$ ). By weak uniqueness, $q_0^{\prime } \upharpoonright N_{i + 1} = q_{i + 1}$ . Since q does not split over $N_i$ , $q \upharpoonright N_0^{\prime }$ does not split over $N_i$ . By monotonicity, also $q_0^{\prime }$ does not split over $N_i$ . By weak uniqueness again (recalling that by definition $N_{i + 1}$ is locally universal over $N_i$ ), $q_0^{\prime } = q \upharpoonright N_0^{\prime }$ . In particular, $q \upharpoonright N_0^{\prime }$ does not split over $M_0$ .⊣

Now since $q \upharpoonright M_1 = p \upharpoonright M_1$ , weak uniqueness and tameness imply that $q \upharpoonright M = p$ , as desired. ⊣

Fact 5.1. Let $\mathfrak {s}$ be a $\text {good}^{-S} \lambda $ -frame on an AEC $\mathbf {K}$ . If $\mathbf {K}$ is $\lambda $ -tame and has weak amalgamation, then $\mathfrak {s}$ is a good $\lambda $ -frame and extends to a good $[\lambda , \infty )$ -frame on all of ${\mathbf {K}}_{\ge \lambda }$ . In particular, ${\mathbf {K}}_{\ge \lambda }$ has amalgamation and arbitrarily large models.

Proof. By [Reference VaseyVas17b, 4.16], ${\mathbf {K}}_{\ge \lambda }$ has amalgamation. By [Reference BoneyBon14], $\mathfrak {s}$ extends to a $\text {good}^{-S} [\lambda , \infty )$ -frame $\mathfrak {t}$ on $\mathbf {K}$ . By [Reference VaseyVas16a, 6.14], $\mathfrak {t}$ , and hence $\mathfrak {s}$ , also has symmetry.Footnote ³ The “in particular” part follows from the definition of a good $[\lambda , \infty )$ -frame. ⊣

Proof. For $N_1 \le _{\mathbf {K}} N_2$ both in ${\mathbf {K}}_\mu $ and $p \in \mathbf {S} (N_2)$ , we say that p does not fork over $N_1$ if there exists $N_1^0 \in {\mathbf {K}}_{<\mu }$ with $N_1^0 \le _{\mathbf {K}} N_1$ such that p does not split over $N_1^0$ . We prove several claims:

• • Nonforking is invariant under isomorphisms, and if $N_1 \le _{\mathbf {K}} N_2 \le _{\mathbf {K}} N_3$ are all in ${\mathbf {K}}_\mu $ and $p \in \mathbf {S} (N_3)$ does not fork over $N_1$ , then $p \upharpoonright N_2$ does not fork over $N_1$ and p does not fork over $N_2$ . This is immediate from the definition and the basic properties of splitting.

• • If $N_1 \le _{\mathbf {K}} N_2$ are both in ${\mathbf {K}}_\mu $ , $p, q \in \mathbf {S} (N_2)$ both do not fork over $N_1$ and $p \upharpoonright N_1 = q \upharpoonright N_1$ , then $p = q$ . To see this use the Löwenheim–Skolem axiom, fix $N_1^0 \le _{\mathbf {K}} N_1$ such that $N_1^0 \in {\mathbf {K}}_{<\mu }$ and both p and q do not split over $N_1^0$ . By Lemma 4.3, all the models in ${\mathbf {K}}_\mu $ are locally model-homogeneous, so in particular $N_1$ is locally universal over $N_1^0$ , so by tameness and weak uniqueness (Lemma 4.7), $p = q$ .

• • If $N \in {\mathbf {K}}_\mu $ and $p \in \mathbf {S} (N)$ , then p does not fork over N. This follows directly from the definition of nice filtrations and Lemma 4.11.

• • If $N_1 \le _{\mathbf {K}} N_2$ are both in ${\mathbf {K}}_\mu $ and $p \in \mathbf {S} (N_1)$ , then there exists $q \in \mathbf {S} (N_2)$ such that q extends p and q does not fork over $N_1$ . Moreover, q can be taken to be nonalgebraic if p is nonalgebraic. To see this, first note that we have observed previously that p does not fork over $N_1$ , hence there is $M_0 \in {\mathbf {K}}_{[\text {LS} (\mathbf {K}), \mu )}$ such that $M_0 \le _{\mathbf {K}} N_1$ and p does not split over $M_0$ . Now we can pick $M_1 \le _{\mathbf {K}} N_1$ with $M_0 \le _{\mathbf {K}} M_1$ , $\|M_0\| < \|M_1\| < \|N_1\|$ , and $\|M_1\|^{\|M_0\|} = \|M_1\|$ , and $\mathbf {K}$ is categorical in $\|M_1\|$ . Then it follows that $M_1$ is locally universal over $M_0$ (see Fact 4.2). Also, Lemma 4.11 implies that $N_1$ is nicely filtrable over $M_1$ . Now apply Lemma 4.12, where $M, N$ there stand for $N_1, N_2$ here.

• • If $\delta < \mu ^+$ is a limit ordinal, $\langle N_i : i \le \delta \rangle $ is increasing continuous in ${\mathbf {K}}_\mu $ , and $p \in \mathbf {S} (N_\delta )$ , then there exists $i < \delta $ such that p does not fork over $N_i$ . Indeed, suppose not. Assume without loss of generality that $\delta $ is regular. In particular, $\delta < \mu $ and so $\delta ^+ < \mu $ . Let $\theta := \text {LS} (\mathbf {K})^+ + \delta ^+$ . Pick $\lambda < \mu $ such that $\mathbf {K}$ is categorical in $\lambda $ and $\lambda ^\theta = \lambda $ . By Fact 4.2, the model in ${\mathbf {K}}_\lambda $ is locally $\theta ^+$ -model-homogeneous. Build $\langle N_i^0 : i < \delta \rangle $ and $\langle N_i^1 : i < \delta \rangle $ increasing in ${\mathbf {K}}_{\lambda }$ such that for all $i < \delta $ , $N_i^0 \le _{\mathbf {K}} N_i$ , $N_i^0 \le _{\mathbf {K}} N_i^1 \le _{\mathbf {K}} N_\delta $ , $N_{i}^1 \cap N_i \subseteq N_{i + 1}^0$ , $p \upharpoonright N_i^1$ splits over $N_i^0$ . This is possible (see [Reference VaseyVas16a, 4.11] for a very similar construction). At the end, let $N_\delta ^\ell := \bigcup _{i < \delta } N_i^\ell $ . Observe that $N_\delta ^1 = N_\delta ^0$ so by Lemma 4.6, there exists $i < \delta $ so that $p \upharpoonright N_\delta ^1$ does not split over $N_i^0$ . This is a contradiction, since we assumed that $p \upharpoonright N_i^1$ splits over $N_i^0$ .

• • For any $N \in {\mathbf {K}}_\mu $ , $|\mathbf {S} (N)| \le \mu $ . Indeed, if $\langle p_i : i < \mu ^+ \rangle $ are types in $\mathbf {S} (N)$ , we can first use Lemma 4.11 to find a nice filtration $\langle N_j : j < \omega \rangle $ of N. In particular, for all $i < \mu ^+$ there exists $j_i < \omega $ with p not splitting over $N_{j_i}$ . By the usual pruning argument (using that $\mu $ is strong limit to see that $|\mathbf {S} (N_j)| < \mu $ for each $j < \mu $ ), there exists $j < \omega $ and an unbounded subset $X \subseteq \mu ^+$ such that for $i_1, i_2 \in X$ , $p_{i_1}$ , $p_{i_2}$ both do not split over $N_j$ and have the same restriction to $N_{j + 1}$ . By weak uniqueness, $p_{i_1} = p_{i_2}$ . This shows that $|\mathbf {S} (N)| \le \mu $ .

Now fix a nonmaximal $M \in {\mathbf {K}}_\mu $ . We identify any $N \in {\mathbf {K}}_\mu $ such that $M \le _{\mathbf {K}} N$ with the corresponding member of ${\mathbf {K}}_M$ (this will not yield to confusion). We define nonforking in the frame using the nonforking relation defined above. The basic types are the nonalgebraic types. We have just established that invariance, monotonicity, uniqueness, extension, and local character hold in the frame. Therefore transitivity and continuity automatically follow (see [Reference ShelahShe09a, II.2.17, II.2.18]). Also note that ${\mathbf {K}}_M$ is not empty (it contains a copy of M) and has no maximal models of cardinality $\mu $ : given $M \le _{\mathbf {K}} N$ in ${\mathbf {K}}_\mu $ , we can fix a nonalgebraic $p \in \mathbf {S} (M)$ (which exists as M is not maximal) and take its nonforking extension to $\mathbf {S} (N)$ . We have observed this extension is not algebraic, hence N cannot be maximal either. We have also seen that $\mathbf {K}$ (hence ${\mathbf {K}}_M$ ) is stable in $\mu $ . Finally, $\mathbf {K}$ has amalgamation in $\mu $ . This follows from Fact 2.5 and the extension property of nonforking. We deduce that ${\mathbf {K}}_M$ has amalgamation in $\mu $ and also joint embedding in $\mu $ .⊣

Proof. Assume without loss of generality that $\mu $ has countable cofinality (otherwise, it is easy to find a smaller cardinal which has countable cofinality and still satisfies the hypotheses). Fix a nonmaximal $M \in {\mathbf {K}}_\mu $ . By Lemma 5.2, there is a $\text {good}^{-S} \mu $ -frame on ${\mathbf {K}}_M$ . It is easy to check that ${\mathbf {K}}_M$ has weak amalgamation and is $\mu $ -tame. Therefore by Fact 5.1, ${\mathbf {K}}_M$ has amalgamation and arbitrarily large model. In particular, $\mathbf {K}$ has arbitrarily large models and M is an amalgamation base in $\mathbf {K}$ . Since any maximal model is an amalgamation base, we deduce that ${\mathbf {K}}_{\ge \mu }$ has amalgamation. ⊣

Fact 5.5 ([Reference Grossberg and VanDierenGV06a, 6.3])

Let $\mathbf {K}$ be an $\text {LS} (\mathbf {K})$ -tame AEC with amalgamation and arbitrarily large models. If $\mathbf {K}$ is categorical in $\text {LS} (\mathbf {K})$ and in $\text {LS} (\mathbf {K})^+$ , then $\mathbf {K}$ is categorical everywhere above $\text {LS} (\mathbf {K})$ .

Proof. Let $\lambda := \text {LS} (\mathbf {K})$ . By [Reference Boney, Grossberg, Vandieren and VaseyBGVV17], $\mathbf {K}$ is $\lambda $ -superstable (essentially, this means that splitting has what is called the $\aleph _0$ -local character for universal chains in [Reference VaseyVas16a]) and by [Reference VaseyVas17a, 5.7(1)], $\mathbf {K}$ has $\lambda $ -symmetry. We now attempt to build a $\text {good}^{-S}$ -frame as in [Reference VaseyVas16a]. We can prove extension without using amalgamation in $\lambda ^+$ , hence by Fact 2.5 we obtain amalgamation in $\lambda ^+$ , and hence obtain a $\text {good}^{-S} \lambda ^+$ -frame. Symmetry is then proven as in [Reference Boney and VaseyBV17b, 6.8] (and is not needed for the present paper).⊣

Proof. By Lemma 5.6, $\mathbf {K}$ has a good $\text {LS} (\mathbf {K})^+$ -frame. By Fact 5.1, it follows that ${\mathbf {K}}_{\ge \text {LS} (\mathbf {K})^+}$ (and hence ${\mathbf {K}}_{\ge \text {LS} (\mathbf {K})}$ ) has amalgamation. Now apply Fact 5.5. ⊣

Proof. Generalizing [Reference ShelahShe09a, I.3.8] (using the corresponding generalization of the Devlin–Shelah weak diamond [Reference Devlin and ShelahDS78] as in [Reference Shelah and VillavecesSV99, 1.2.4]), we can find $\lambda \in [\text {LS} (\mathbf {K}), \theta )$ such that ${\mathbf {K}}_\lambda $ has amalgamation. Now apply Theorem 5.7 to ${\mathbf {K}}_{\ge \lambda }$ . ⊣

Proof. Let $\mu := \beth _\omega (\text {LS} (\mathbf {K}))$ . By Theorem 3.6, not every element of ${\mathbf {K}}_\mu $ is maximal. By Theorem 5.4, $\mathbf {K}$ has arbitrarily large models. Now apply Corollary 5.8. ⊣

Proof. Let $\phi $ be a universal $\mathbb {L}_{\omega _1, \omega }$ sentence and let $\mathbf {K}$ be its the class of models (ordered with substructure). Note that $\mathbf {K}$ has intersections (the closure operator computed inside N is the closure under the functions of N). By Lemma 2.8, $\mathbf {K}$ has coherent sequences. By Remark 2.4, $\mathbf {K}$ has weak amalgamation. By [Reference VaseyVas17b, 3.7], $\mathbf {K}$ is $\text {LS} (\mathbf {K})$ -tame. Let $\chi \in [\aleph _0, \beth _\omega )$ be such that $\mathbf {K}$ is categorical in every cardinal in $[\chi , \beth _\omega )$ . Now apply Corollary 5.9 to ${\mathbf {K}}_{\ge \chi }$ . ⊣

Proof. By results of Shelah on building good frames, there is a good $\lambda $ -frame on $\mathbf {K}$ and ${\mathbf {K}}_{\le \lambda ^+}$ is $\lambda $ -tame (see [Reference VaseyVas20, 7.1] for an outline of the proof). By Fact 5.1, $\mathbf {K}$ has arbitrarily large models. Now apply Corollary 5.8. ⊣