Membership of 6×6 Hoffman–Pereira matrices in the Parrilo hierarchy

Determine whether every 6×6 matrix satisfying the Hoffman–Pereira sign condition belongs to some cone K_6^{(r)}, or, equivalently, whether the existence of Hoffman–Pereira matrices outside \bigcup_{r\geq 0}K_6^{(r)} remains possible.

Background

The paper studies sum-of-squares hierarchies $K_n^{(r)}$ and $Q_n^{(r)}$ as inner approximations to the copositive cone. For Hoffman–Pereira matrices—matrices with unit diagonal and off-diagonal entries in $\{-1,0,1\}$ satisfying the triangle-free and distance-two sign conditions—the authors establish certificates in a more flexible hierarchy $\widetilde{Q}_n^{(r)}$.

The authors construct a 7×7 Hoffman–Pereira matrix, namely the direct sum of the Horn matrix and a 2×2 matrix, that belongs to no $K_7^{(r)}$, and extend this obstruction to every dimension $n>7$ by taking direct sums with identity matrices. Their argument therefore leaves the dimension $n=6$ unclassified: it does not determine whether a corresponding 6×6 obstruction exists or whether all 6×6 Hoffman–Pereira matrices eventually enter the $K_6^{(r)}$ hierarchy.