Is the decision whether a $k$-clique exists in a $k$-partite graph NP-hard?
I have found only a very limited number of references on this problem, and they seem to be concerned with heuristics to enumerate the cliques (in particular k-cliques in k-partite graphs). On complexity, they only comment that the max-clique problem is generally hard, but nothing on the specific case.
Note: This is an edit of my earlier question: whether the max-clique problem in a $k$-partite graph is NP-hard, with $k$ being part of the input? As Austin pointed out in the comments, it is easy to see that the answer is trivially yes by a reduction from the general max-clique problem; any graph $G$ on $n$ vertices can be considered $n$-partite. The new question, however, is more specific and a reduction does not seem so obvious. For example, (and contrary to the original question) for $k=n$ one can easily check if an $n$-partite graph contains/is an $n$-clique. What about general $k$?