6 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…

Boney–VanDieren's interval conjecture. The set of regular cardinals $\mu$ such that the $(\lambda,\mu)$-limit model is isomorphic to the $(\lambda,\lambda)$-limit model might be an…

Let $\mathbb{K}$ be an Abstract Elementary Class and let $\lambda>\text{LS}(\mathbb{K})$ be such that $\mathbb{K}_\lambda$ has amalgamation and is Galois stable in $\lambda$. Consi…

VanDieren's conjecture. The set of regular ordinals $\alpha<\lambda^+$ such that the $(\lambda,\alpha)$-limit model is isomorphic to the $(\lambda,\lambda)$-limit model is an end s…

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…