26 problems

An Abstract Elementary Class is a class of structures equipped with a strong substructure relation; $\text{LS}(d53e)$ denotes its Löwenheim–Skolem number, and $d53e$ is categorical…

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…

Categoricity-above-the-Hanf-number conjecture. If $\mathcal{K}$ is categorical above the Hanf number, then $\mathcal{K}$ has the $\mu$-amalgamation property for every $\mu$.

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. Say that $\operatorname{\mathcal{K}}$ is categorical in a cardinal when it has exactly one model of that cardinali…

Let $\operatorname{\mathcal{K}}$ be an abstract elementary class. If $\operatorname{\mathcal{K}}$ is $\chi$-tame for some cardinal $\chi$, then there exists a cardinal…

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…

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…

Baldwin's transitivity conjecture. Transitivity of nonsplitting holds for models that are universal over one another.

Grossberg's conjecture. If

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…

Let $\mathbf{K}$ be an abstract elementary class, let $\lambda$ be a cardinal, and suppose that $\mathbf{K}$ has amalgamation in cardinality $\lambda^+$. A good $\lambda$-frame is…

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 $\mathcal{K}$ be an abstract elementary class and let $\mu$ be a cardinal. The relevant notions are uniqueness of limit models of cardinality $\mu$ and superstability for…

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 an intermediate logic mean a logic stronger than ${\mathbb L}_{\lambda^+,\aleph_0}$ but weaker than ${\mathbb L}_{\lambda,\lambda}$, with the stronger demand that Ehrenfeucht–M…

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 $K_T$…