In a group of 6 people either we have 3 mutual friends or 3 mutual enemies. In a room of n people?

That's a reasonable answer. I think my favorite way of viewing this question is to think of a complete graph on six vertices (i.e. each vertex is connected to each other vertex), where all edges are colored either red or blue. Then you are to show that there is either a red triangle or a blue triangle.

You could put your reasoning on this picture, as it becomes very easy to follow.

This is also a good time to first learn about Ramsey Theory - of which this is one of the easiest examples. Look here.