Forking Degree and Generic Transitivity

Definitions and mathematical statement

The supplied record presents five related research questions, rather than a single fully specified conjecture. Throughout, $\alpha$ denotes an arbitrary ordinal, including a possibly infinite ordinal, and $k$ denotes a positive integer.

For terminology concerning primitivity, a definable action of a group $G$ on a nonempty definable set $X$ is called definably primitive if it is transitive and every definable $G$-invariant equivalence relation on $X$ is either equality or the universal relation. Here invariance means that, for every $g\in G$ and $x,y\in X$, \[ E(x,y)\quad\Longleftrightarrow\quad E(g\cdot x,g\cdot y). \] The ambient structure and the parameters permitted in the word "definable" must be specified in any precise application; the supplied passage does not specify them.

The passage does not define $\alpha$-forking degree, its version of generic $k$-transitivity, its generalized transitivity conditions, or the particular Borovik--Cherlin conjecture to be extended. Nor does it specify the binding-group constructions or the differential settings needed for its Kolchin-polynomial questions. These terms are retained as source terminology, without imposing additional hypotheses or choosing a particular formulation.

The requested research problems are as follows.

1. For arbitrary ordinals $\alpha$, can $\alpha$-forking degree be controlled using an appropriate version of the Borovik--Cherlin conjecture? The intended degree, the objects to which it applies, the controlling quantities, and the form of the bound remain unspecified.
2. Generalizations of generic $k$-transitivity have been studied for algebraic group actions. Find natural model-theoretic applications of these stronger or more general variants of the Borovik--Cherlin setup. The passage does not identify the variants or prescribe a particular application.
3. Formulate and prove a version of the Borovik--Cherlin conjecture for supersimple groups, and use it, through $\mathrm{ACFA}$, to obtain functional transcendence results for solutions of difference equations. Here $\mathrm{ACFA}$ is the standard notation for the theory of existentially closed difference fields, interpreting the passage's typography "ACF A." The intended supersimple-group hypotheses, characteristic, difference equations, transcendence quantities, and inequalities are not supplied.
4. The passage reports that the functional transcendence applications in its references [6--8] admit a reduction in which any binding group associated with the bounds can be assumed to be definably primitive. Determine whether definable primitivity of such binding groups can be related to currently studied notions in geometric stability theory, differential algebraic geometry, or the theory of compact complex manifolds. The references, bounds, binding groups, candidate notions, and intended meaning of a relation are not identified in the supplied record.
5. Formulate and prove versions of the Borovik--Cherlin conjecture for Kolchin polynomials in various settings. The differential structures, relevant Kolchin polynomials, comparison relations, and proposed bounds must be specified as part of such formulations.

These questions retain their source-dependent qualifications. The supplied evidence does not determine a unique quantified theorem or conjecture for any of the requested extensions.

Short English statement

This is a five-part research programme concerning ordinal forking degrees, generalized generic transitivity, supersimple groups and difference equations, definably primitive binding groups, and Kolchin polynomials. Several definitions and target bounds are absent, so the questions are preserved with their unresolved qualifications. The supplied open-status label is a dated curation claim, not an independently verified mathematical conclusion.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30005284$, title "Forking Degree and Generic Transitivity," in the Oberwolfach Reports Open Problems collection. Exact source problem code: catalogue entry. No individual proposer is supplied.
• Supplied report citation: Mini-Workshop: Topological and Differential Expansions of o-minimal Structures (2023), Oberwolfach Reports, pp. 3005--3050, DOI 10.4171/owr/2022/52. Supplied source and literature-source link, labelled "EMS Press, OWR 2022/52": https://doi.org/10.4171/owr/2022/52. The record gives proposed year 2022, while its report title contains 2023; this discrepancy is unresolved.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the questions open and reports that no complete solution of the listed generalizations was found. This is supplied dated curation, not an independently verified literature assessment. The background expressly requests independent verification of time-sensitive status and cited claims.
• The record labels its statement corrected_verified and says that it was reconstructed and checked against the official report. These are supplied verification claims; no independent source visit or verification is asserted here. The passage cites references [6--8], but their bibliographic entries are absent from the supplied record.