The vertex-minor Ramsey number conjecture for five vertices

Let $R_{\mathrm{vm}}(k)$ denote the smallest integer $n$ such that every graph on $n$ vertices contains the edgeless graph $E_k$ as a vertex-minor. The known values for $k=2,3,4$ are $3$, $7$, and $11$, respectively, and the next unresolved case concerns $k=5$.

Vertex-minor Ramsey number conjecture for $k=5$.

$$R_{\mathrm{vm}}(5)=15.$$

This is the next unresolved case in the study of vertex-minor Ramsey numbers. It extends the observed pattern $R_{\mathrm{vm}}(k)=4k-5$ for $k=2,3,4$, while the general bounds allow substantially larger values.