Prove that finding largest subset of undirected graph that is almost independent is NP-hard
The proof is by reduction from independent set. Given an instance $(G,k)$ of independent set, construct a new instance $(G',1000k)$ in the following way: each vertex $x \in G$ is replaced by an independent set $I_x$ of size 1000, and each edge $(x,y)$ is replaced by a complete bipartite graph $I_x \times I_y$.

If $G$ contains an independent set $S$ of size $k$, then $G'$ contains an independent set $\bigcup_{x \in S} I_x$ of size $1000k$. In the other direction, suppose that $G'$ contains an almost independent set $S$ of size $1000k$. Remove all vertices incident to any edges within $S$ to obtain an independent set $S' \subseteq S$ of size at least $1000k-200$. Let $T = \{ x : I_x \cap S' \neq \emptyset \}$, and note that $T$ is an independent set of $G$. Finally, since $S'$ can contain at most 1000 vertices from each $I_x$, we see that $|T| \geq \lceil |S'|/1000 \rceil = k$.