Let $G$ be a "hypergraph", a collection of vertices $V=\{v_1,v_2,\ldots,v_n\}$ and a collection of "hyperedges" $E=\{e_1,e_2,\ldots,e_m\}$, where $e_i\subseteq V$ and unlike normal edges, an edge may contain more than two vertices.
An "independent set" (http://en.wikipedia.org/wiki/Independent_set_(graph_theory)) is a collection of vertices, $U$, that does not fully contain any of the hyperedges: $e_i\not\subseteq U$. The "independence number" or "maximum independent set size" is the size of the largest independent set in the graph $G$.
I know that finding if there is an independent set of size $k\in\mathbb{N}$ in some normal graph $G$ is NP-Complete. If I am not mistaken, this holds for hypergraphs as well. However calculating the independence number is not proven to be NP. Even approximating it is not proven in NP.
First, is there a more specific complexity class for calculating the independence number than NP-Hard?
Second, how much harder is it for a hypergraph? Again, is there a complexity class more specific?
For related questions, a recent dissertation has been helpful to me: https://escholarship.org/uc/item/79t9b162.
Thanks!