Facet-gap problem for order and chain polytopes

For every finite poset $P$ with $n=|P|$ , determine the facet-count gap $\operatorname{gap}(P):=f_{n-1}(C(P))-f_{n-1}(O(P))$ , where $O(P)$ and $C(P)$ are respectively the order and chain polytopes of $P$ . In particular, seek a structural determination, classification, and sharp bounds for $\operatorname{gap}(P)$ beyond the known inequality $f_{n-1}(O(P))\leq f_{n-1}(C(P))$ . The analogous problem for the facet differences between admissible decompositions of marked chain--order polytopes is also included.

References

Primary source

arXiv

Additional references

On facet gaps of order and chain polytopes — arXiv — Ghislain Fourier

combinatorics polyhedral-geometry

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed progress

A new 2026 paper gives a local counting rule for the difference between two polytope constructions, but the general problem remains open.

The problem concerns the facet-count difference between order and chain polytopes of a finite poset, known to satisfy $f_{n-1}(O(P))\le f_{n-1}(C(P))$ . The broader question asks how this gap can be structurally determined, including for marked chain–order polytopes.

Known results

The gap is bounded using crossing numbers associated with elements of maximal antichains.
Gap zero is characterized by avoidance of a specified $X$ -shaped subposet.
Posets with $\operatorname{gap}(P)=1$ are classified.
Classification for $\operatorname{gap}(P)=2$ and sharper general bounds remain open.

August 2026 local-weight advance

Ghislain Fourier’s paper reports that local weights determine facet differences for finite posets and for all admissible decompositions of marked chain–order polytopes. It advances several earlier questions but explicitly does not resolve every facet-gap problem.

Current status (as of August 2026): Local-weight formulas are reported for finite posets and admissible marked chain–order decompositions, while broader facet-gap classifications and a complete resolution remain open.

Sources

No solutions have been posted yet.