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