Matroid Structure and Hyperpaths in Collapsible Hypertrees

Definitions and mathematical statement

For a finite abstract simplicial complex $K$, an elementary simplicial collapse removes a pair of nonempty faces $\sigma\subsetneq\tau$, where $\dim\tau=\dim\sigma+1$ and $\tau$ is the unique maximal face containing $\sigma$. The complex is collapsible if a finite sequence of elementary collapses reduces it to a single vertex. This specifies the usual simplicial meaning of collapsibility; the supplied record does not explicitly specify finiteness or its convention for collapse.

A matroid on a finite ground set $E$ is a family $\mathcal I\subseteq 2^E$ such that $\varnothing\in\mathcal I$, every subset of a member of $\mathcal I$ belongs to $\mathcal I$, and whenever $I,J\in\mathcal I$ satisfy $|I|<|J|$, there exists $e\in J\setminus I$ with $I\cup\{e\}\in\mathcal I$. The source's phrase "matroid-like behavior" does not specify that these exact axioms must hold, nor does it identify a ground set or an associated family of independent sets or bases.

Research question as supplied. Do collapsible $d$-dimensional hypertrees, also called $d$-trees in the original wording, exhibit useful matroid-like behavior? Is there a satisfactory notion of a $d$-hyperpath, described in the edited statement as a higher-dimensional analogue of a path for these hypertrees?

Unresolved formulation. The supplied record does not define a $d$-dimensional hypertree, specify the range of $d$, give vertex-set or finiteness assumptions, or state any required homological conditions and coefficient domain. It also gives no axioms for "useful" matroid-like behavior or a "satisfactory" hyperpath. In particular, the original wording does not explicitly restrict the hyperpath question to collapsible hypertrees, whereas the edited wording suggests such a restriction. These questions therefore remain a research program rather than a fully specified quantified conjecture; no particular hypertree convention, matroid construction, or hyperpath definition is asserted here.

Short English statement

The source asks whether collapsible higher-dimensional trees have useful matroid-like structure and whether an appropriate higher-dimensional notion of a path can be developed. It leaves the definitions, parameter ranges, and success criteria unspecified, so a precise yes-or-no conjecture requires further source clarification.

Sources

• Local UnsolvedMath registry, supplied record with permanent ID $30004219$, title "Matroid Structure and Hyperpaths in Collapsible Hypertrees," source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record gives proposed year 2019 and names no proposer. Its status is supplied as "open." Its statement-status label is corrected_verified, with an explanation that notation, typography, and sentence structure were repaired while preserving mathematical content; this is supplied editorial metadata, not independent mathematical verification.
• Supplied report citation: Geometric, Algebraic, and Topological Combinatorics (2020), Oberwolfach Reports, pp. 2395--2472, DOI 10.4171/owr/2019/39. The literature-source label is "Oberwolfach Report 39, hypertrees." https://doi.org/10.4171/owr/2019/39. The background attributes the problem collection to workshop open-problem discussions documented in Oberwolfach Reports; no exact problem page or speaker attribution is supplied.
• Supplied literature reference: "Dahari and Linial, hyperpaths." https://doi.org/10.1007/s00454-021-00360-x. No full title or further bibliographic details are supplied.
• Literature assessment checked at 2026-08-22: the supplied curation labels the problem open and states that sum complexes provide infinite collapsible and non-collapsible families, while broader matroid-like behavior of collapsible hypertrees and a satisfactory $d$-hyperpath theory remain open. This is supplied dated curation derived from the OWR material, not an independently verified assessment. The source expressly calls for independent verification of time-sensitive status and cited claims.