BALKAN 1989 Problem 4

The elements of the set $F$ are some subsets of $\left\{1,2,\ldots ,n\right\}$ and satisfy the conditions: i) if $A$ belongs to $F$ , then $A$ has three elements; ii)if $A$ and $B$ are distinct elements of $F$ , then $A$ and $B$ have at most one common element. Let $f(n)$ be the greatest possible number of elements of $F$ . Prove that $\frac{n^{2}-4n}{6}\leq f(n) \leq \frac{n^{2}-n}{6}$