Finite regular hypertopes for arbitrary finite-label Coxeter diagrams

Determine whether every Coxeter diagram whose edge labels are all finite admits a finite regular hypertope with that diagram.

Background

The paper studies twisting constructions for coset incidence systems and regular hypertopes. These constructions preserve finiteness under suitable hypotheses and yield finite regular hypertopes for certain families of Coxeter diagrams, notably trees whose edge labels are all equal to 4 except possibly one. This motivates the broader unresolved question of whether finiteness of all edge labels alone is sufficient for the existence of a finite regular hypertope realizing a given Coxeter diagram.

The question is not resolved in general by the paper; the results provide only a positive answer for the specified tree-shaped family. If an edge label is infinite, the corresponding regular hypertope must be infinite because it contains two involutions whose product has infinite order.

References

Indeed, one can ask the question: “Let $\mathcal{D}$ be a Coxeter diagram whose labels are all finite. Does there exists a finite regular hypertope whose diagram is $\mathcal{D}$?”

— Coset geometries acting on their elements: The $j$-diagonals twisting  (2608.23223 - Piedade et al., 24 Aug 2026) in Introduction