Optimal vertex thresholds for pathological rigidity matroid constructions

Determine the smallest vertex thresholds for which the constructions in Theorem 1 and its corollaries produce symmetric abstract d-rigidity matroids distinct from the standard rigidity, hyperconnectivity, and cofactor matroids.

Background

The main theorem establishes the construction for n ≥ d + 11, while the principal corollary uses n ≥ 2d + 9. The authors explicitly state that these bounds have not been optimized and identify several remaining small cases in dimension two, so determining the sharp thresholds is left unresolved.

References

The remaining cases to check for $d=2$ are for $n \in {8,9,11}$.

— Constructing pathological symmetric abstract rigidity matroids  (2610.12036 - Dewar et al., 8 Oct 2026) in Section 4, “Open problems,” subsection “Extending main results to hyperconnectivity and cofactor matroids”

We conjecture that \Cref{maincor2} is true in all dimensions higher than 1.

— Constructing pathological symmetric abstract rigidity matroids  (2610.12036 - Dewar et al., 8 Oct 2026) in Section 4, “Open problems,” subsection “Extending main results to hyperconnectivity and cofactor matroids,” displayed Conjecture