20 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 $E$…

Let $adN$ denote the nonstandard natural numbers, let $qen$ be the equivalence relation used in the source, and let $Exp\dN$ be the exponential equivalence relation. A Borel equiva…

Let $\boe\res\boldsymbol{N}$ denote the relation on $\boldsymbol{N}$ defined by $x\boe y$ if there is a Borel bijection from $[0,x)$ onto $[0,y)$. Let…

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…

Zapletal's conjecture. Under these assumptions,

We discuss the Borel cardinality $\|T_{\overline{M}}\|$ of the theory $T_{\overline{M}}$, and write $\beth_2$ for the lower bound established for tame expansions in the surrounding…

In Generalized Descriptive Set Theory, consider the isomorphism relations of classifiable and non-classifiable theories on structures of the relevant uncountable cardinality. Fried…

Hjorth–Kechris–Louveau strictness conjecture. The following Borel reducibilities are strict:

Orbit-equivalence versus idealistic-equivalence conjecture. Orbit equivalence relations coincide with the idealistic equivalence relations.

The analytic non-orbit-equivalence conjecture. The equivalence $E_1$ is not the least analytic equivalence relation which is not Borel reducible to an orbit equivalence.

For each $n\geq 1$, let $\cong_n$ be the isomorphism relation on the space of torsion-free abelian groups of rank $n$, and let $\cong_{n+1}^+$ be the corresponding relation with th…

For each $n\geq 1$, let $\mathcal{BD}^*_n$ be the class of Bratteli diagrams of exact rank $n$, and let $BD^*_n$ be the equivalence relation induced by equivalence of Bratteli diag…

Let $\bigsqcup_{n\geq 1} \cong_n^+$ denote the disjoint union of the isomorphism relations $\cong_n^+$ on the corresponding spaces of torsion-free abelian groups, and let…

Let $\mathfrak{M}_\mathsf{Sep}$ be the space of separable complete metric spaces, and let $\cong^i$ denote isometry on this space. Recall that the isometry relation for zero-dimens…

Törnquist's conjecture. The isomorphism relation for separably acting factors is universal, with respect to Borel reducibility, for orbit equivalence relations induced by a continu…

Gao's conjecture. Isometry between zero-dimensional Polish metric spaces is Borel bireducible with any complete orbit equivalence relation.

Thomas's conjecture. The resulting orbit equivalence relations are pairwise incomparable with respect to Borel reducibility.

Thomas's conjecture. These orbit equivalence relations are neither Borel reducible to one another nor stably orbit equivalent.

Dichotomy conjecture. Every reasonable Borel set of reductions induces either a Lipschitz-like or a Wadge-like hierarchy of degrees.

Let $m,n\geq3$ and let $p,q$ be primes. Write $\oiso_{m,p}$ for isomorphism equivalence on the space of $p$-local torsion-free abelian groups of rank $m$, and $\oqiso_{n,q}$ for qu…