Full-Dimension Intersections in Definable Families

Definitions and mathematical statement

The supplied record concerns a definable family $(X_a)_{a\in A}$ of subsets of a pseudofinite set $Y$. Here $A$ is the parameter set indexing the family, and $\delta$ denotes the dimension appearing in the source. The ambient structure, allowed parameters for definability, precise meaning of pseudofiniteness, and definition and codomain of $\delta$ are not supplied. They must be specified before the following target becomes a precise mathematical problem.

The cleaned formulation assumes \[ X_a\subseteq Y,\qquad \delta(X_a)=\delta(Y)\quad\text{for every }a\in A. \] A definable subset $B\subseteq Y$ has full $\delta$-dimension, in the terminology of this formulation, when $\delta(B)=\delta(Y)$. The requested intersection property is \[ \delta(X_a\cap B)=\delta(X_a)+\delta(B)-\delta(Y). \] The cleaned question asks for hypotheses under which a generic parameter $a$ has this property for every definable full-dimensional $B\subseteq Y$. This is a request to identify hypotheses, not a supplied theorem or a fully specified conjecture. The dimension arithmetic in the displayed equality must itself be defined; in particular, no convention for subtracting infinite dimension values is supplied.

There are unresolved qualifications. The original fragment assumes $\delta(X)=\delta(X_a)$, with $X$ undefined, whereas the cleaned formulation assumes $\delta(X_a)=\delta(Y)$ for every $a\in A$. Identifying $X$ with $Y$ is not justified by the supplied fragment. The original fragment also does not mention pseudofiniteness, generic parameters, or the displayed intersection formula. These are features of the cleaned formulation whose provenance needs review. Moreover, neither genericity nor the permitted parameters defining $B$ are specified. It is unclear whether one generic parameter must work simultaneously for every eligible $B$, or whether genericity may depend on a previously fixed $B$. Thus the record does not determine a unique quantified mathematical statement.

Short English statement

The cleaned question asks when a generic member of a definable family of full-dimensional subsets meets every definable full-dimensional subset with the stated dimension equality. The dimension, genericity, parameter restrictions, and some hypotheses are unspecified, and the original fragment differs from the cleaned version. This is an unresolved statement-only draft requiring human mathematical review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003021$, title Full-Dimension Intersections in Definable Families, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record assigns the year 2016; no individual proposer is supplied.
• Supplied report citation: Model Theory: groups, geometry, and combinatorics (2016), Oberwolfach Reports, pp. 5--50, DOI 10.4171/owr/2016/1. Source URL and supplied literature-source link, labelled "Model Theory report": https://doi.org/10.4171/owr/2016/1. The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. No exact problem location within the cited page range is supplied.
• Supplied literature assessment dated 2026-08-21: the extracted fragment gives no definitions beyond an equality of $\delta$-dimensions and no precise conclusion to test; it is described as unresolved and normalized to open. This is supplied dated curation, not an independently verified literature assessment. The record expressly calls for independent verification of time-sensitive status and cited claims.
• The registry separately labels the statement corrected_verified and reports that it was checked against the official Oberwolfach report and preserves its hypotheses, notation, and open target. This is a supplied verification claim, not verification performed for this draft; its relationship to the underdetermined original fragment and the dated assessment requires review.