18 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…

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

Let $\operatorname{\mathcal{K}}$ be an abstract elementary class (AEC), and let … Assume that $\operatorname{\mathcal{K}}$ has at least one model but fewer than the maximal number…

Let ${\bf K}$ be an abstract elementary class and let $\lambda=\operatorname{LS}({\bf K})$, its Löwenheim–Skolem number. Shelah's Hanf-number threshold conjecture. The threshold fo…

Let $\psi$ be a sentence of $L_{\lambda^+,\omega}$ in a language of size $\lambda$. Categoricity means having exactly one model, up to isomorphism, in a given cardinality. Shelah's…

Grossberg's conjecture. If

Let $\lambda$ be a cardinal. For an abstract elementary class (AEC) $\boldsymbol{K}$, write $\operatorname{LS}(\boldsymbol{K})$ for its Löwenheim–Skolem number, and say that…

Let $\theta$ be a function assigning a cardinal $\theta(\tau)$ to each cardinal $\tau$. If $d4e6$ is an abstract elementary class categorical in some…

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

Grossberg–VanDieren's categoricity-to-tameness conjecture. Suppose $K$ is an AEC. If $K$ is categorical in some $\lambda \geq Hanf(LS(K))$ (or some other value depending only on…

Let ${\mathfrak k}$ be an abstract elementary class and put $\kappa={\rm LST}_{\mathfrak k}+|\tau_{\mathfrak k}|$. Superlimit-or-nonstructure conjecture. At least one of the follow…

Let ${\mathfrak k}$ be an abstract elementary class. Large-cardinal dichotomy conjecture. For a closed unbounded class $C$ of cardinals, the alternatives stated in the source shoul…

Stationarity dichotomy conjecture. The classes $S_1$ and $S_2$ are not both stationary. A weaker version replaces $S_2$ by the class of $\lambda\in C^{\rm fp}_{\aleph_0}$ such that…

Let ${\mathfrak k}$ be an abstract elementary class. Main gap and super-limit conjecture. (1) The main gap theorem holds for $K_{\mathfrak k}$ for all sufficiently large $\lambda$.…

Let ${\mathfrak k}=(K_{\mathfrak k},\leq_{\mathfrak k})$ be an abstract elementary class, with Löwenheim–Skolem–Tarski number ${\rm LST}_{\mathfrak k}$. Categoricity transfer conje…

Let ${\mathbb P}$ be a c.c.c. forcing notion of cardinality $\lambda$ such that forcing with ${\mathbb P}$ forces Martin's axiom and $2^{\aleph_0}=\lambda$, where…

Let $T$ be a countable first-order theory and let $K_T$ be its class of atomic models. Excellence conjecture. If $K_T$ is categorical in every $\beth_n$ for $n<\beth_0$, then $K_T$…

Let $T$ be a countable first-order theory and let $K_T$ be its class of atomic models. Baldwin's conjecture. If $K_T$ is categorical in $\beth_1$, then $K_T$ is $\beth_0$-stable, e…