408 problems

For every $Z$-group $G$ and every normal subgroup $N\trianglelefteq G$, is $N$ again a $Z$-group? Equivalently, is the class of all $Z$-groups closed under taking normal subgroups?

Let $T$ be a countable complete theory and let $P$ be a distinguished unary predicate in its vocabulary. Assume that $T$ fails the Gaifman property; that is, assume that there exis…

Every locally nilpotent countably categorical group is nilpotent; equivalently, for every group $G$, if $G$ is locally nilpotent and countably categorical, then $G$ is nilpotent.

Let $\mathcal T_1$ be a classifiable theory and $\mathcal T_2$ a non-classifiable theory. A generalized Borel-reducibility main gap problem. asks whether there is a Borel reduction…

Let $\underline{M}$ be the class of orders generated from the one-point order by finite sums, products with $\omega$ and $\omega^*$, and the specified dense rational-indexed sums.…

Shelah's three cardinal and model-existence claims. The following assertions hold:

Petrykowski's conjecture. If $G$ has a bounded orbit, then $G$ is definably amenable.

Characterizability conjecture.

Let $G$ be an infinite simple tame $K^*$-group, where a $K^*$-group is a group of finite Morley rank in which every proper definable subgroup is a $K$-group, and a $K$-group is one…

Let ${\mathcal T}_{\infty}$ be the theory under consideration, and let $\sim$ be a definable equivalence relation over ${\mathcal T}_{\infty}$. The relation is finite if its equiva…

Assume that $T$ is a first-order theory and that … with $2^\lambda=\lambda^+$. Let $\langle M_\alpha:\alpha<\lambda^+\rangle$ be a $\prec$-increasing continuous sequence with…

Let $M$ be a first-order topological structure, and let $\operatorname{Spec}(A)$ denote the Stone spectrum of a definable set $A\subseteq M^k$. Say that $M$ is definably $T_5$ when…

Let $M\preccurlyeq N$, let $X=\operatorname{Spec}(A)$ be the o-minimal spectrum under consideration, let $Z\subseteq X$ be definably closed, and let $W:=\theta^{-1}(Z)$, where…

Let $K$ be a $p$-adically closed field and let $A$ be an infinite definable subset of $K^n$ that is open in its closure. Let $(B_k)_{k\leq q}$ be a finite collection of closed defi…

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 $M$ be a countable recursively saturated model of $\mathrm{PA}$, and let $\operatorname{SSy}(M)$ denote its standard system. Transcendence characterization conjecture. There is…

Characterization conjecture. The group $G$ is elementarily equivalent to a non-abelian free group if and only if $G$ is a non-elementary hyperbolic fully residually free tower.

Small-index conjecture. If $H$ is a subgroup of $Autf(M)$ with index strictly less than $2^{\aleph_{0}}$, then there exists a finite subset $A\subset M$ such that

Let $T$ be a stable theory, let $M$ be a model of $T$, and let $A\subseteq M$ be arbitrary. The set $A$ is weakly benign in $M$ if equality of strong types over $A$ implies equalit…

Let $T$ be a stable theory, let $M$ be a model of $T$, and let $A\subseteq M$. The set $A$ is benign in $M$ if equality of types over $A$ implies equality of the corresponding type…

Open-domain projective categoricity conjecture. Two such sets $U$ and $V$ are $\mathcal{L}_{\mathrm{B}}$-elementarily equivalent if and only if they are projectively equivalent.

Ball categoricity conjecture. $K\equiv_{\mathcal{L}_{\mathrm{B}}}\overline{B}{}^{\,n}$ if and only if $\partial K$ is a quadric, equivalently, if and only if $K$ is projectively eq…

Projective categoricity conjecture. $K\equiv_{\mathcal{L}_{\mathrm{B}}}K'$ if and only if $K$ and $K'$ are projectively equivalent.

Stable forking conjecture over a base. If $a \mathrel{\mathop{\mathpalette\notind{}}_{C}} b$, there is a formula $\varphi(x,b)\in\operatorname{tp}(a/Cb)$ which forks over $C$, such…

Stable forking conjecture. If $a \mathrel{\mathop{\mathpalette\notind{}}_{C}} b$, then there is a formula $\varphi(x,\overline{b})\in\operatorname{tp}(a/Cb)$ such that…