3 problems

Let $T$ be a complete countable theory, let $\mathbf{s}$ be a suitable sequence, and let $\mathcal{U}_{\mathbf{s}}$ be the associated ultrafilter. Write $k_*(\mathbf{s})$ for the s…

Let $T$ be a simple theory and let $3 \leq k \leq \aleph_0$. Consider the following conditions: (A) $T$ has $\mathcal{P}^-(k)$-amalgamation of models; (B) $T$ has $<k$-type amalgam…

Let $EC$ be an elementary class with the strict order property $SOP_2$, meaning that there are a model $M\in EC$, a first-order formula $\varphi(x;y)$, and parameters…