11 problems

The maximality conjecture. The relation \mathrel{\raise0.2ex\text{\ooalign{\hidewidthvert\hidewidth\cr\raise-0.9ex\text{smile}}}}^{nm} is the maximum strongly admissible geom…

Let $\mathbb{G}$ be a locally compact Polish group, and call a subset of $\mathbb{G}$ topologically universal if every dense $G_\delta$ subset of $\mathbb{G}$ contains its image un…

Let $G$ be a Polish group, and let $C\subseteq G$ be a comeagre conjugacy class, meaning a conjugacy class whose complement is meagre. Say that $G$ has the countable index property…

Let $G$ be a Polish group. Say that $G$ has the countable index property when every subgroup of index at most countable is open, and say that $G$ has the automatic continuity prope…

Sabok's conjecture. The action of $\operatorname{Aut}(X,\mu)$ on itself by conjugation is a complete orbit equivalence relation.

Let $\mathbb{G}$ be a Polish group. A nonempty subset $C\subseteq\mathbb{G}$ is anti-GMS if it is nowhere dense and, for every sequence $\{U_n:n\in\omega\}$ of open neighborhoods o…

Let $\mathbb{G}$ be a Polish group. A set $S\subseteq\mathbb{G}$ is of strong measure zero, and a set $M\subseteq\mathbb{G}$ is meager. Galvin–Mycielski–Solovay characterization co…

Ding–Gao–Hjorth conjecture. If $E\leq_B E^X_G$ and $E$ is essentially countable, then $E$ is essentially hyperfinite.

Essential hyperfiniteness conjecture. If

Orbit-relation conjecture. Every $G$-orbit equivalence relation is Borel reducible to

Gurarij space conjecture. The Gurarij space $\mathbf{G}$ has WEMI, and every transitive complete $\operatorname{Aut}(\mathbf{G})$-space is a singleton.