3 problems

Let $K_n$ and $K_m$ be complete graphs on $n$ and $m$ vertices, respectively, and let $K_n\Box K_m$ denote their Cartesian product. Write $mr_{+}^{\mathbb{R}}(K_n\Box K_m)$ for its…

Let $C_m$ be the cycle graph on $m$ vertices and $P_n$ the path graph on $n$ vertices. Their Cartesian product $C_m\Box P_n$ has minimum positive semidefinite rank…

Let $S_n$ be the slack matrix of the regular $n$-gon, and let $\operatorname{rank}_{\operatorname{psd}}(S_n)$ denote its positive semidefinite rank. Logarithmic psd-rank conjecture…