Let $V$ be the vertices of the cycle graph $C_{3n}$. Suppose there is a partition of $V$ into sets of $3$, i.e. $V=\cup_{k=1}^{n}{V_k}$ where $|V_k|=3$ for $k$ in $1..n$.

**QUESTION:** Is it possible to find an independent set of $V$ with exactly one vertex from each $V_k$?

By the Lovasz Local Lemma, it's possible if the $3$ is replaced with some larger number, say, $11$.