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 $T$ be a complete theory with quantifier elimination in a relational language $L$, and let $P$ be a distinguished unary predicate symbol in $L$. For a model $M\models T$, write…

Let $T$ be a countable complete first-order theory with a distinguished predicate $P$. Say that $T$ is categorical over $P$ when it has at most one model, up to isomorphism over…

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…

Let $\mathcal{T}$ be the class of $\aleph_0$-categorical, $\aleph_0$-stable classical theories in countable languages, and let $\mathcal{D}\subseteq\mathcal{T}$ consist of those th…

Let $Q$ be an $\aleph_0$-categorical, $\aleph_0$-stable theory in a countable language, and let $Q_P$ denote the theory of beautiful pairs of models of $Q$. Approximate categoricit…

Grossberg's conjecture. If

Let $\mathcal{A}$ be a nice accessible category with directed colimits having just one model of presentability rank $\kappa$. Weaker conjecture. Then $\mathcal{A}$ must be categori…

Eventual categoricity conjecture. There is a threshold cardinal $ctheta$ such that if $cpsi$ is categorical in some $cmu cge ctheta$, then $cpsi$ is categorical in all…

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

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…

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 ${\mathfrak k}=(K_{\mathfrak k},\leq_{\mathfrak k})$ 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…