Is k -rainbow coloring of a hypergraph NP-complete or not?
Deciding if a hypergraph can be rainbow colored in $k$ colors is NP-complete for every $k \geq 2$. A proof is given in [1, Theorem 12].

For $k=2$, the result follows immediately by observing the problem is equivalent to set splitting (also known as hypergraph 2-coloring). For $k \geq 3$, the authors of [1] give a reduction from chromatic index on $k$-regular graphs.



[1] M. Koivisto, P. Laakkonen, and J. Lauri. "NP-completeness results for partitioning a graph into total dominating sets". In Proceedings of the 23rd Annual International Computing and Combinatorics Conference (COCOON), volume 10392 of Lecture Notes in Computer Science, pages 333-345. Springer, 2017.