Triangle-free graph with 5 vertices What is the maximum number of edges in a triangle-free graph on 5 vertices?
No answers, please...just hints.
I believe that E $\leq$ 5, but I'm not sure where to go from there.

It is not true that this is $5$. Connect one point to three points, and connect the remaining point to the same three points. 
To proof this optimal you could start distinguishing cases by maximum degree of a vertex. There might be a more elegant argument yet I would not know it momentarily.
