3 problems

Let $\operatorname{\mathcal{K}}$ be an abstract elementary class, let $\operatorname{Hanf}(\operatorname{\mathcal{K}})$ and $LS(\operatorname{\mathcal{K}})$ denote its Hanf and Löw…

Interval conjecture. This set is a non-trivial interval of regular cardinals. Moreover, its minimum is an important measure of the complexity of $\mathcal{K}$.

Let $T$ be dependent, let $|T|<\lambda=\lambda^{<\lambda}$, let $\lambda^+=2^\lambda$, and let $\sigma=\operatorname{cf}(\sigma)<\lambda$. Suppose…