244 problems

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…

Weiss's conjecture. Every orbit equivalence relation induced by a Borel action of a countable amenable group is hyperfinite.

Let $X$ be an ordinal-definable set of reals, and let $\mathbb{P}$ denote Jensen's “minimal $\Pi^1_2$ real singleton forcing”. Let $\mathbb{P}^{\omega}_{\mathrm{fs}}$ be the finite…

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 $2^\omega$ be the Cantor group under coordinatewise addition, and call $X\subseteq 2^\omega$ strongly meager if, for every measure zero set $H$, one has … Dual Borel Conjecture…

Let $E$ be a Borel equivalence relation, and let $E_1$ denote eventual agreement on countable sequences of reals. For equivalence relations $E$ and $F$, write $E\leq_B F$ when $E$…

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…

Sabok's conjecture. The conjugacy relation of minimal compact systems is a complete orbit equivalence relation.

For each $n\in\omega$, let $b_{2n+1,0}$ denote the cardinal invariant defined in the source from the direct-limit images of the relevant iterates of $M_{2n+1}(x)$, and let…

Let $n\in\omega$. A cardinal is $\bf{\Gamma}$-reachable if there is a sequence of distinct $\bf{\Gamma}$-sets of that length, and is $\bf{\Gamma}$-unreachable otherwise. Kechris pr…

A countable Borel equivalence relation is an equivalence relation on a Borel subset of $2^\omega$ whose equivalence classes are countable and whose relation is Borel. A Borel reduc…

Let $\textup{\textsf{NIN}}$ denote the problem of finding a function witnessing non-injectivity, and let $\textup{\textsf{HBU}}$ denote the Heine–Borel theorem for covers of…

Under $AD^+$, let an ordinal definable real mean a real definable from ordinal parameters, and let a real appear in an $d714_1$-iterable mouse mean that it belongs to such a mouse.…

Debs–Saint Raymond conjecture. For every analytic ideal $\mathcal{I}$ and $0<\alpha<\omega_1$,

Let $F$ be a Borel equivalence relation, let $\mathbb E_1$ denote eventual agreement on sequences of reals, and let $E_G^X$ denote the orbit equivalence relation of a group action.…

Consistency-strength dichotomy conjecture. The consistency strength of this statement is always either ZFC or ZFC together with the existence of an inaccessible cardinal; moreover,…

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…

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…

Blackwell determinacy conjecture. The game $\Gamma(S)$ should be determined, that is,

Galvin's type conjecture. For every Borel coloring of $n$-tuples of reals, there is a perfect set $P$ such that the color of every increasing $n$-tuple from $P$ depends only on its…

Least non-Polish-action conjecture. The relation $Ei$ is a $\rebs$-least equivalence relation in the non-Polish-action domain.

Sufficiency conjecture. c93-chain-condition universal Baireness suffices for the argument above, namely for proving that $0^{\#}$ exists.

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 $\mathbb{R}$, let $A\subseteq X$ be Borel, let $\mathcal{A}$ denote the ideal of Aronszajn-null sets, and let $\mathcal{A}_{\mathsf{G}}$ be…

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…