Erdős Problem #1079 — A dense neighborhood at the Turán threshold

For $r≥4$, let $f_r(n)$ be the least number of edges forcing a $K_r$ in every $n$-vertex graph. Is there a constant $c_r>0$ such that every $n$-vertex graph with $f_r(n)$ edges has a vertex of degree $m>c_rn$ whose neighborhood spans at least $f_{r-1}(m)$ edges?