Idealistic Equivalence Relations and Orbit Relations

Definitions and mathematical statement

A standard Borel space is a measurable space isomorphic to the Borel measurable space of a Polish space. For equivalence relations $E$ on a standard Borel space $X$ and $F$ on a standard Borel space $Y$, a Borel reduction of $E$ to $F$ is a Borel map $f:X\to Y$ satisfying \[ \forall x,x'\in X\qquad x\,E\,x'\iff f(x)\,F\,f(x'). \] Write $E\leq_B F$ when such a map exists, and $E\sim_B F$ when both $E\leq_B F$ and $F\leq_B E$.

Under the usual descriptive set theoretic convention, an orbit equivalence relation is obtained from a Polish group $G$ acting on a standard Borel space $Y$ through a Borel action map $a:G\times Y\to Y$. Here $a(1_G,y)=y$ and $a(gh,y)=a(g,a(h,y))$. The associated relation is \[ y\,E_a\,y'\iff (\exists g\in G)\;a(g,y)=y'. \] The supplied record does not specify this acting-group and action convention, so its agreement with the source must be checked.

The supplied record also does not define idealistic. Let $\operatorname{Ideal}_{\mathrm{src}}(E)$ stand for the intended source property; this is an unresolved predicate, not a replacement definition. In particular, no additional assumption that $E$ is a Borel subset of $X\times X$ is inserted.

Subject to resolving that predicate and confirming the orbit convention, the question is whether \[ \begin{gathered} \text{for every standard Borel space }X\text{ and every equivalence relation }E\text{ on }X,\\ \operatorname{Ideal}_{\mathrm{src}}(E)\ \Longrightarrow\ \exists\,Y,G,a,f,h\quad \left\{ \begin{aligned} &Y\text{ is a standard Borel space},\quad G\text{ is a Polish group},\\ &a:G\times Y\to Y\text{ is a Borel action},\\ &f:X\to Y\text{ and }h:Y\to X\text{ are Borel},\\ &\forall x,x'\in X\quad x\,E\,x'\iff f(x)\,E_a\,f(x'),\\ &\forall y,y'\in Y\quad y\,E_a\,y'\iff h(y)\,E\,h(y'). \end{aligned} \right. \end{gathered} \] Thus the requested conclusion is $E\sim_B E_a$. The missing definition of idealistic prevents this draft from being a fully self-contained formulation. The supplied source labels the question open; that status is not independently verified here.

Short English statement

The question asks whether every idealistic equivalence relation on a standard Borel space has exactly the same complexity under Borel reducibility as some orbit equivalence relation. The supplied record leaves the definition of idealistic and the precise orbit-action convention unstated; these require source confirmation.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30006165$, title Idealistic Equivalence Relations and Orbit Relations, exact source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection.
• Set Theory (2025), Oberwolfach Reports, DOI 10.4171/owr/2025/2, pp. 75--140. Supplied official-report reference: https://doi.org/10.4171/owr/2025/2. The excerpt attributes Idealistic equivalence relations remastered to Filippo Calderoni, joint work with Luca Motto Ros. The record gives 2025 as the proposed year but leaves its separate proposer field empty.
• Calderoni--Motto Ros, Structural results on idealistic equivalence relations, supplied manuscript link: https://sites.math.rutgers.edu/~fc327/idealistic.pdf.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the question open and reports that the 2026 primary manuscript still describes the general Borel bireducibility question as open, while proving a strong obstruction for a related classwise embedding formulation. This is supplied dated curation, not an independently verified assessment; its time-sensitive status and cited claims require independent checking.
• The original excerpt mentions the reference label [KecLou], without supplying its bibliographic entry. Its identity and relevance remain unresolved.
• The registry labels the statement corrected_verified and says it was reconstructed and checked against the official Oberwolfach report. This records the supplied verification claim; no source was independently visited or checked for this draft.