On the stability of graph independence number

Let $G$ be a graph on $n$ vertices of independence number $\alpha (G)$ such that every induced subgraph of $G$ on $n-k$ vertices has an independent set of size at least $\alpha (G)-\mathrm{\ell }$. What is the largest possible $\alpha (G)$ in terms of $n$ for fixed $k$ and $\mathrm{\ell }$? We show that $\alpha (G)\le n\mathrm{/}2+C_{k,\mathrm{\ell }}$, which is sharp for $k-\mathrm{\ell }\le 2$. We also use this result to determine new values of the Erd\H{o}s–Rogers function.