Monotonicity of RCB and RRCB

Prove whether the rectangle covering bound and refined rectangle covering bound are monotone as the number of vertices increases for regular n-gon slack matrices and linear Euclidean distance matrices.

Background

The authors identify a theoretical question concerning whether both combinatorial bounds increase monotonically with the parameter n for two families of matrices: slack matrices of regular n-gons and linear Euclidean distance matrices. Although monotonicity appears intuitive, the paper states that it has not been established.

References

More interesting from a theoretical standpoint is the question: Are the RCB and RRCB monotone regarding the $n$ of regular $n$-gons or LEDMs? This seems intuitive, but to the best of our knowledge, this has not been proven yet.

— Computing Lower Bounds on the Non-negative Rank via SAT Solvers  (2610.11950 - Engel et al., 8 Oct 2026) in Section 6, Conclusion, paragraph “Future Work”