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, $11$.



Restricted independent set of the cycle graph $C_{3n}$
The answer is yes, via a stronger proposition: for each part in the partition, add edges between the vertices the part contains; the question now is, can you vertex-3-color the resulting graph? This is the well-known *cycle plus triangles problem*, originally posed by Erdos and solved by Fleischner and Stiebitz using the Alon-Tarsi theorem, and later again by Sachs (inductively), affirmatively. The proofs are far from trivial or easy (I happen to like the inductive proof). 


So to answer your question, just pick a color class ... 


