122 problems

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…

An ordinal definable game of length $\omega_1$ on natural numbers with real parameters is a game in which the payoff and defining data are ordinal definable, with real parameters a…

Let $G_{\operatorname{cont}}(\nu,A)$ be the game of continuously coded length, where $\nu$ is a partial function from reals to natural numbers and $A$ is a payoff set of sequences…

Assume ${\sf{AD}}^+$. Let $\kappa$ be a regular Suslin cardinal, and let ${\mathbf{\Gamma}}$ be the pointclass of $\kappa$-Suslin sets. Suslin-cardinal unreachable successor conjec…

Let ${\mathbf{\delta}}^1_{2n+2}$ denote the supremum of the lengths of ${\mathbf{\Delta}}^1_{2n+2}$ pre-well-orderings of ${\mathbb{R}}$. Assume ${\sf{ZF+AD+DC}}$. Kechris's second…

Let ${\mathbf{\delta}}^1_{2n+2}$ denote the supremum of the lengths of ${\mathbf{\Delta}}^1_{2n+2}$ pre-well-orderings of ${\mathbb{R}}$. Assume ${\sf{ZF+AD+DC}}$. Kechris's first…

Let $X$ be a separable Banach space that is not isomorphic to a Hilbert space. Recall that $X$ is ergodic if the equivalence relation $\mathbf{E}_0$ is Borel-reducible to the isomo…

Let $A_{(\alpha,n)}$ be the sets defined from $J_{\alpha}(\mathbb{R})$, and let a mouse set mean a set $A\subseteq\mathbb{R}$ of the form $A=\mathbb{R}\cap\mathcal{M}$ for some cou…

Let $X$ be a separable Banach space, let $A\subseteq X$ be Borel, and let $\mathbb{A}_{\mathsf{G}}$ be the auxiliary game referred to in the paper, with $\mathbb{A}$ denoting the c…

Let $X$ be a separable Banach space over $cmathbb{R}$, let $A\toolbox X$ be Borel and Aronszajn-null, and let $\mathcal{A}_{\mathsf{G}}$ be the game in which player $\mathbf{I}$ ch…

Let $X$ be a Banach space with a basis, respectively an unconditional basis, which is not equivalent to the canonical basis of $c_0$ or $\ell_p$ for $1\leq p<+\infty$. A normalized…

Let $X$ be a Banach space with an unconditional basis. A block-subspace is a closed subspace generated by a block-sequence of that basis, and $E_0$ is the equivalence relation of e…

A separable Banach space is a Banach space with a countable dense subset, and a Banach space is ergodic when the isomorphism relation between its infinite-dimensional subspaces is…

Let $X$ be a compact topological space, and suppose every closed subset of $X$ is a $G_\delta$ set, meaning a countable intersection of open subsets of $X$. Fremlin's two-to-one co…

Let $I$ be a $c3$-ideal on $2^\omega$ or $\omega^\omega$, and let $P_I$ be the corresponding factor poset. Assume that $P_I$ is proper and has continuous reading of names. Trace-id…

A MAD family is a maximal almost disjoint family of infinite subsets of $\text{omega}$, and a subset $X\subseteq 2^\omega$ is a sigma-set if for every Borel set…

Let $G$ be a Polish group and let $X$ be a Polish $G$-space, meaning that $G$ acts continuously on $X$. An orbit is a set of the form $G\cdot x$ for some $x\in X$; two points are e…

Let $X$ be a set, let $\operatorname{\mathscr{L}}$ be a language, and let $\kappa$ be the relevant cardinal. Let $\mu_0$ and $\mu_1$ be invariant measures on…

Mathias's question. In Solovay's model, are there infinite MAD families? More generally, is it a theorem of $\mathsf{ZF}+\mathsf{DC}_{\mathbb R}$ that there are no infinite MAD fam…

Let $T\subseteq \mathbb{Y}$ be a subtree, and write … For $d>\frac{1}{2}$ and $\delta\in\left[\frac{1}{2},d\right]$, consider win-lose alternating-move games on $T$ with payoff set…

Let $G$ be a connected Lie group, let $ath.E(G,\mathcal{D}(G))$ denote the conjugacy relation on the space $\mathcal{D}(G)$ of discrete subgroups of $G$, and call an equivalence re…

Let $Q=(q_n)_{n\geq 1}$ be a nondeterministic dynamically generated basic sequence. Let $\mathcal{N}(Q)$, $\mathcal{DN}(Q)$, and $\mathcal{UN}(Q)$ denote the corresponding normalit…

Let ${\mathsf{AD}}$ denote the axiom of determinacy and let ${\mathsf{Bl}\text{-}\mathsf{AD}}$ denote the axiom of Blackwell determinacy. Martin's conjecture.…

Let EMPTs denote ergodic measure-preserving transformations, and consider their isomorphism relation under measure-preserving conjugacy. An equivalence relation is Borel reducible…

Let $E$ be a Borel equivalence relation, and let $E_1$ denote eventual agreement on countable sequences of reals. Write $E\leq_B F$ when $E$ is Borel reducible to $F$. An orbit equ…