Neighbors of the uniform-matroid vertex

Characterize or describe the matroids, and respectively the isomorphism classes of matroids, that yield a vertex neighboring the uniform-matroid point in \overline{\Omega}_{r,n} and \Omega_{r,n}.

Background

The paper singles out adjacency to the uniform matroid as a special case of the general edge problem. In the labelled setting the reference point is \overline{p}{U{r,n}}, while in the unlabelled setting it is the corresponding point for the uniform matroid; identifying its neighboring vertices would give concrete information about the edge structure.

References

Characterize or describe the matroids (resp.\ isomorphism classes of matroids) that yield a vertex neighboring the point $\overline{p}{U{r,n}$ in $\overline{\Omega}{r,n}$ (resp. $p{U_{k,n}$ in $\Omega_{r,n}$).

— The polytope of all matroids  (2502.20157 - Ferroni et al., 27 Feb 2025) in Section 6.3, “Edges of the polytopes”