Minimal Ordered Structures Beyond Almost Linearity

Definitions and mathematical statement

In standard model-theoretic terminology, an infinite structure $\mathcal M$ with universe $M$ is called minimal if every subset of $M$ definable in $\mathcal M$, with parameters from $M$, is finite or cofinite. Here cofinite means that its complement in $M$ is finite. The supplied record does not explicitly specify its convention for minimality, so this standard convention requires confirmation against the cited report.

The supplied existence question is: does there exist a minimal ordered structure that is not almost linear? The distinguished group case asks whether such a structure can be a minimal ordered group.

More formally, the intended questions have the schematic forms \[ \exists\mathcal M\;\left(\mathcal M\text{ is minimal and ordered in the source's sense, and is not almost linear in the source's sense}\right) \] and, for the group case, \[ \exists\mathcal G\;\left(\mathcal G\text{ is a minimal ordered group in the source's sense, and is not almost linear in the source's sense}\right). \] These are existential questions; the group case imposes an additional requirement on the witness. The supplied record gives no definition of "ordered" or "almost linear." In particular, it does not specify the order axioms, whether the order is named or definable with parameters, its compatibility with a group operation, or the precise almost-linearity condition. These source-dependent predicates remain unresolved, so the displays are a faithful statement schema rather than a complete self-contained formalization.

Short English statement

Does a minimal ordered structure exist that is not almost linear? Can such an example be an ordered group? The source-specific meanings of "ordered" and "almost linear," and the convention for minimality, require confirmation. The supplied record labels the problem partially solved; this draft does not independently verify that status.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30002263$, title "Minimal Ordered Structures Beyond Almost Linearity," source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record associates the question with 2013 and names no individual proposer.
• Supplied source citation: Model Theory: Groups, Geometry, and Combinatorics (2013), Oberwolfach Reports, pp. 5--66, DOI 10.4171/owr/2013/01. https://doi.org/10.4171/owr/2013/01. The registry marks the statement "exact" and reports that the official report explicitly states this existence question and retains it as open after the almost-linear classification. These are supplied source-verification claims, not independently verified here.
• Supplied literature reference: Around Podewski's Conjecture. https://doi.org/10.4064/fm222-2-4.
• Supplied literature reference: Model Theory, Oberwolfach Report 01/2013. https://doi.org/10.4171/owr/2013/01.
• Literature assessment dated 2026-08-21: the supplied curation labels the problem partially solved and states that the almost-linear case is classified, that no almost-linear minimal field exists, and that existence of a minimal ordered structure that is not almost linear remains unresolved. This is supplied dated curation derived from the OWR collection, not an independently verified literature assessment; its time-sensitive status and cited claims require independent verification.