Let $k\ge 2$ and $G$ be a graph with $n\ge k-1$ vertices and$$(k-1)(n-k+2)+\left(\genfrac{}{}{0ex}{}{k-2}{2}\right)+1$$edges. Does there exist some $c_k>0$ such that $G$ must contain an induced subgraph on at most $(1-c_k)n$ vertices with minimum degree at least $k$?