124 problems

Let $G=(X,E)$ be a locally finite Borel graph on a standard Borel space $X$. Does there always exist a Borel coloring $c:X\to\{0,1\}$ such that, for every $x\in X$, the number of n…

Let $A$ and $B$ be simple separable $C^*$-algebras, and let $\widehat{A}$ and $\widehat{B}$ denote their spectra, namely the spaces of unitary-equivalence classes of irreducible un…

Martin's conjecture. Under these assumptions: (I) if $f$ is degree-invariant and is not increasing a.e., then $f$ is constant a.e.; and (II) $\leq_m$ pre-well-orders the set of deg…

Let $T$ be an elementary first-order theory, and consider its countable models up to isomorphism. Vaught's conjecture. The number of countable models of $T$ is either countable or…

Let $2^{\omega}$ be Cantor space. A function $f\colon 2^{\omega}\to 2^{\omega}$ is $(\equiv_T,\equiv_m)$-invariant if Turing-equivalent inputs have many-one-equivalent outputs; it…

Let $(W_i)_{i\in\mathbb{N}}$ be an effective enumeration of the computably enumerable sets, and write $W_i\approx W_j$ when the two sets lie in the same orbit under automorphisms o…

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…