Smaller subgraphs of minimum degree k

In 1990 Erd\H{o}s, Faudree, Rousseau and Schelp proved that for $k\ge 2$, every graph with $n\ge k+1$ vertices and $(k-1)(n-k+2)+\left(\genfrac{}{}{0px}{}{k-2}{2}\right)+1$ edges contains a subgraph of minimum degree $k$ on at most $n-\sqrt{n}\mathrm{/}\sqrt{6k^3}$ vertices. They conjectured that it is possible to remove at least $\epsilon_k n$manyverticesandremainwithasubgraphofminimumdegree$k$,forsome$\epsilon_k>0$. We make progress towards their conjecture by showing that one can remove at least $\mathrm{\Omega }(n\mathrm{/}\log n)$ many vertices.