First we see that each vertex $v$ has degree at least $27$ else there exists 3 vertices not connected with $v$ and thus 4 non connected subgraph. So we have $$\varepsilon \geq {30 \cdot 27 \over 2} = 405$$ This value is achivable if we take $G$ as of $10$-partite graph with $3$ vertices in each partition.