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