Finishing degree requirements NP-complete: Choosing 1 element from each set avoiding conflicts
Assuming the sets $C_i$ are pairwise disjoint, this problem is very related to the independent set problem and is called maximum colored independent set (also called rainbow colored independent set). The problem goes as follows. Given a graph and a colouring function over the vertices of the graph. Find a maximum independent set with each vertex in the set having a different colour. The problem is NP-hard as well (try to reduce Independent set).

Assuming the sets are not disjoint the problem does not get easier since the version with disjoint sets is only a special case of this problem.