Vertex-minor Ramsey numbers: exact values and extremal structure

We determine the vertex-minor Ramsey number $\text{Rvm}(4)=11$, where $\text{Rvm}(k)$ is the smallest~$n$ such that every $n$-vertex graph contains the edgeless graph~$E_k$ as a vertex-minor. We prove this by an exhaustive classification of the graphs on~$10$ and~$11$ vertices under local complementation. At the extremal order $n=10$, exactly six non-isomorphic graphs avoid~$E_4$ as a vertex-minor; up to isomorphism, they represent five LC-equivalence classes, and each labeled LC orbit has cardinality~$\mathrm{8,712}$. Thus $k=4$ is the first case in which the general upper bound $2^k-1$ is not attained. Using the extremal graphs as building blocks, we derive explicit lower bounds on~$\text{Rvm}(k)$ that surpass the leading term of the asymptotic bound for all $k\le 9$; in particular, $\text{Rvm}(5)\ge 13$. We also describe structural properties of the six extremal graphs and formulate the next open problem, whether $\text{Rvm}(5)=15$.