31 problems

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 $E$…

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 $F$ 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.…

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

Let $2^{\mathbb N}$ be Cantor space, and let $x\equiv_T y$ mean that $x$ and $y$ have the same Turing degree. Regard $(\equiv_T)$ as a countable Borel equivalence relation under Bo…

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…

Zapletal's conjecture. Under these assumptions,

Spectrum conjecture. For each countable ordinal $\alpha$, $=^{+\alpha}$ is in the spectrum of the meager ideal.

Countable invariant uniformization conjecture. Let $E$ be a Borel equivalence relation on a Polish space $X$. Then the following are equivalent: (a) $E$ is reducible to countable;…

Let $E_1$ be the eventual-equality equivalence relation on $\{0,1\}^{{\mathbb N}\times{\mathbb N}}$: for sequences $\langle x_n\rangle_n$ and $\langle y_m\rangle_m$ of elements of…

Let $\mathbb E_1$ denote eventual agreement on sequences of reals, and let $F$ be a treeable Borel equivalence relation. Say that $F$ is essentially countable when it is Borel-redu…

Let $\mathbb E_1$ be the equivalence relation on sequences of reals defined by eventual agreement, and let a Borel equivalence relation be prime if it is Borel-reducible to no prod…

A countable Borel equivalence relation is an equivalence relation on a standard Borel space whose classes are countable and which is Borel as a subset of the product space. Marks's…

Measure-reducibility conjecture. If $E$ and $F$ are countable equivalence relations on standard Borel spaces and $E$ is measure reducible to $F$, then $E$ is reducible to $F$.

Nonuniversality conjecture. Recursive isomorphism on $2^\omega$ is not a universal countable Borel equivalence relation.

Treeable nonreducibility conjecture. The universal treeable countable Borel equivalence relation is not Borel reducible to Turing equivalence.

Optimistic uniform universality conjecture. If $E$ is a universal countable Borel equivalence relation, then $E$ is uniformly universal with respect to every way of generating $E$.

Nonuniversality conjecture. Recursive isomorphism on $2^\omega$ is not a universal countable Borel equivalence relation.

Uniform universality characterization. A countable Borel equivalence relation is universal if and only if it is uniformly universal with respect to every way it can be generated.

Let $E$ be a Borel equivalence relation with infinitely many $E$-classes. Write $E^{\mathrm{fin}}$ for the equivalence relation on finite subsets that identifies two finite subsets…

Essential hyperfiniteness conjecture. If

Let $\overline a$ and $\overline b$ be countable tuples from a monster model, with $\overline b$ containing $\overline a$. Let $E_L^{\overline a}$ and $E_L^{\overline b}$ be the co…

Let $T$ be a theory with monster model ${\EuFrak C}$, let $\overline{a}$ be a countable tuple from ${\EuFrak C}$, and let $E_L^{\overline a}$ and $F_L^{\overline a}$ denote respect…

Let $G$ be a group definable or $*$-definable over $\emptyset$ in a monster model, and let $E_G$ be the associated Borel equivalence relation coding the quotient $G^{00}/G^{000}$.…

Let $T$ be a first-order theory, and let $Gal_0(T)$ denote the kernel of the canonical map from the Lascar group $Gal_L(T)$ to the Kim--Pillay group $Gal_{KP}(T)$. Lascar-group non…