Let $L$ be a context-free language, $\bar L$ be its complement and $\bar L_n$ be the length $n$ words in $\bar L_n$.
What is known about $|\bar L_n|$?
Note that it is known that $|L_n|$ is either polynomial (if $L$ is bounded$~$), or grows exponentially. I wonder if anything similar might be true about $|\bar L_n|$. Warning, $L$ is allowed to be ambiguous!