Labeled Floor Diagrams in Three Dimensions

Definitions and mathematical statement

The supplied normalized statement concerns tropical curves in three-dimensional projective space and their labeled floor diagrams. The supplied original question is: "What can be said about the three-dimensional case, especially about the number of labeled floor diagrams?" Neither a definition of a three-dimensional floor diagram nor its labeling convention is included in the supplied record.

Statement of the requested investigation. Develop a precise combinatorial description and enumeration of labeled floor diagrams associated with tropical curves in three-dimensional projective space.

To express the intended enumeration schematically, write $\mathcal{D}^{\mathrm{lab}}_3(\lambda)$ for the collection of diagrams after all defining conventions and enumerative parameters $\lambda$ have been specified. The requested count would then be \[ \#\mathcal{D}^{\mathrm{lab}}_3(\lambda). \] Here $\mathcal{D}^{\mathrm{lab}}_3(\lambda)$ is only a placeholder for the missing definition, not a collection determined by the supplied evidence. The record does not specify the degree, genus, incidence conditions, admissible graph data, label constraints, or equivalence relation. It also does not specify the ambient tropical projective model, boundary conditions, or whether the intended enumeration is an ordinary cardinality or a count with multiplicities. No finiteness assertion or quantification over these parameters can therefore be extracted reliably. The original wording alone does not identify the ambient projective space; that interpretation comes from the supplied normalized statement. A fully specified enumerative problem requires recovery and mathematical review of the relevant source context.

Short English statement

The problem asks for a combinatorial description and enumeration of labeled floor diagrams for tropical curves in three-dimensional projective space. The supplied excerpt omits the diagram definition, labeling rules, and enumerative parameters, so it does not determine a unique counting problem. The supplied status is partially solved; this is a dated curation label requiring independent review.

Sources

• Local UnsolvedMath registry, data/unsolvedmath/problems.json, permanent ID $30002784$, title "Labeled Floor Diagrams in Three Dimensions," source problem code catalogue entry. The supplied record places the problem in the Oberwolfach Reports Open Problems collection and records the proposed year as 2015. No individual proposer is identified.
• Supplied primary-source citation: Geometric and Algebraic Combinatorics (2015), Oberwolfach Reports, pp. 285--368, DOI 10.4171/owr/2015/5. https://doi.org/10.4171/owr/2015/5. The background describes these reports as documenting mathematical workshops and their open-problem discussions. The supplied literature list also identifies this link as the "original OWR report."
• Supplied literature reference, labeled "Rau, counting tropical rational space curves": https://arxiv.org/abs/2003.06174.
• Supplied literature assessment, checked on 2026-08-21: status "partially solved." The assessment states that floor-diagram methods give correspondence and enumeration results for important tropical space-curve problems in three dimensions, while no general resolution of the broad requested enumeration was verified in that assessment. This is supplied dated curation derived from the OWR material, not an independently verified literature assessment; its time-sensitive status and cited claims require independent verification.
• The registry assigns statement status corrected_verified and reports that the official report passage was checked directly and its question, definitions, and notation restored without strengthening the claim. This is supplied verification metadata, not independent verification by this draft; the complete definitions and notation are absent from the supplied record.