SOLVED

Let $k\geq 0$. Let $G$ be a graph on $n$ vertices such that every subgraph $H\subseteq G$ contains an independent set of size $\geq \frac{1}{2}\lvert H\rvert-k$. Must $G$ be the union of a bipartite graph and $O_k(1)$ many vertices?

Proved by Reed [Re99]. (Thanks also to Reed for pointing out that the case $k=0$ is trivial, since if $G$ is not bipartite then $G$ contains an odd cycle.)