Consider a graph with $n$ vertices such that if one takes any 4 vertices there are at most 4 edges among these 4 vertices (Notice that there are 6 "possible" edges among these 4 vertices). What is the maximum possible edge density for such a graph as a function of $n$ and what is the limit as $n$ goes to infinity?

Using a probabilistic argument one can show that as $n$ goes to infinity that the edge density can be at most $\frac{2}{3}+\epsilon$, yet finding such a construction seems extremely tricky.