Shelah's three cardinal and model-existence claims

Let $\alpha$ be an ordinal, and let $\chi<\mu<\lambda<\operatorname{Ded}^*\chi$. For a countable theory $T$, let a $\lambda$-like model mean a model of cardinality $\lambda$ in the sense used in the source. Let $\psi\in L_{\omega_1,\omega}$.

Shelah's three cardinal and model-existence claims. The following assertions hold:

A)

$$(\aleph_{\alpha+\omega+\omega},\aleph_{\alpha+\omega},\aleph_\alpha)\rightarrow(\lambda,\mu,\chi).$$

B) If $T$ has a $\lambda$-like model, $\lambda$ is a limit cardinal, and $|T|\leq\mu<\lambda_1\operatorname{Ded}^*(\mu)$ for a singular cardinal $\lambda_1$, then $T$ has a $\lambda_1$-like model. If $\lambda$ is an $\omega$-Mahlo weakly inaccessible cardinal, the stated cardinal and singularity restrictions on $\lambda_1$ can be removed.

C) If $\psi$ has a model of cardinality $\aleph_{\omega_1}$, then it has a model of cardinality $2^{\aleph_0}$.

These are two-cardinal, partition, and model-existence assertions from Shelah's work. The source flags the last assertion with a reference to [Sh:522], so its status and the precise interpretation of the notation should be checked against that reference.