Large induced subgraphs with $k$ vertices of maximum degree

We prove that, for every integer $k\ge 2$, there exists a constant $c_k>0$ such that every graph on $n\ge R(k,k)$ vertices with maximum degree $\mathrm{\Delta }$ contains an induced subgraph on at least $n-c_k\sqrt{\mathrm{\Delta }}$ vertices whose maximum degree is attained by at least $k$ vertices. This confirms a conjecture of Caro and Yuster in strong form.