# Disagreement with professor over NP reduction problem

I’m a slight disagreement with my professor over whether or not a certain reduction is possible. He asked us to reduce the problem of 3-Coloring to the problem of 3-Clique. The problem is that I’m fairly confident that 3-Coloring is NP-Complete, while 3-Clique is P. Correct me if I’m wrong (which is very likely), but for any k-clique where the k is fixed, is $$V^k$$, meaning the 3-clique is $$V^3$$, right? I asked my professor about this and his response was:

“3-clique is definitely not in P. You (apparently) have to examine all thrices of vertices to settle the matter.”

And I still don’t understand how this is not a $$V^3$$ operation.

If I figured out a way to reduce 3-coloring to 3-clique wouldn’t I be millionaire?