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.

Consider a pentagon. If you try to add any more edges to the pentagon a triangle will be formed. Thus for graph having a cycle containing 5 vertices ( all vertices that is ) can have at maximum 5 edges without violating the condition.
Now consider bi-partite graphs with a total of 5 vertices , say $x$ in one group and $5-x$ in other. Bipartite graphs can't have odd length cycles. Thus no triangles and no pentagons.
As we have already taken care of pentagons. You just need to find maximum in case of bi-partite graph. In bi-partite graph maximum edges can be $(5-x)*x$. Hope I have not answered completely.
