20 problems

Let $\mathbb{G}$ be a Polish group, and consider the Galvin–Mycielski–Solovay property that strong measure zero sets are characterized by avoiding translates or products with every…

Hjorth's reduction conjecture. Under these hypotheses, every such analytic orbit quotient can be represented as the orbit quotient of an analytic subset of a Polish $S_{\infty}$-sp…

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…

Separation conjecture. Most of these definitions should be separable from one another by direct relativisation of known effective results or by using Pontryagin duality and the cor…

For a Polish group $\mathbb{G}$, define … Also let $\operatorname{non}(\operatorname{Smz}(\mathbb{G}))$ be the least cardinality of a subset of $\mathbb{G}$ that is not of strong m…

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.

Polish Direct Summand Conjecture. 1. If $G$ has a direct summand isomorphic to $G^*_{s,\lambda}$ for some $\aleph_0<\lambda\leqslant 2^{\aleph_0}$ and $s\in S_*$, then it has one o…

Topological Vaught Conjecture. For every Polish $G$-space $(\mathcal{X},G,\cdot)$, the orbit equivalence relation $E_G$ has either countably many or perfectly many classes.

Assume that $M$ is a countable, saturated model of a small, NIP theory in a countable language, and let $\Delta$ be a finite set of formulas without parameters. A Polish structure…

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.

Non-hyperfinite full-group conjecture. If $\mathcal R$ is not hyperfinite, then $[\mathcal R]$ does not have ample generics.

Let $G$ be a Polish group, and let $M(G)$ denote its universal minimal flow. A $G$-flow has a generic orbit if it contains a comeager orbit; $G$ has the generic point property when…

$\mathcal{N\!M}$-gap conjecture. The $\mathcal{N\!M}$-rank of every orbit in a small Polish structure is either a finite ordinal or $\infty$.