Let $M_n(\mathbb{R})$ denote the $n^2$-dimensional real vector space of real $n\times n$ matrices. Let $\rho_k(n)$ denote the maximum dimension of a subspace $V$ of $M_n(\mathbb{R})$ such that every nonzero matrix in $V$ has at least $k$ nonzero eigenvalues. A famous result of Adams (see https://pdfs.semanticscholar.org/c88c/235fc5386b2ba0bb347c3a2d20f2d905d3e5.pdf) states that $\rho_n(n)=2^c+8d$, where $n=(2a+1)2^b$ and $b=c+4d$, and where $a,b,c,d$ are integers with $0\leq c<4$. Moreover, it is easy to see that $\rho_1(n) = {n+1\choose 2}$. Namely, the value is achieved by the symmetric matrices. Let $W$ be the ${n\choose 2}$-dimensional subspace of strictly upper-triangular matrices. Every element of $W$ is nilpotent, i.e., has only 0 eigenvalues. Any subspace of dimension greater than ${n+1\choose 2}$ will intersect $W$ in at least a one-dimensional subspace, proving that $\rho_1(n)={n+1\choose 2}$.

What is known about $\rho_k(n)$ for $1<k<n$?