Let $S$ be a set of size $v$ and let $T$ be a set of $3$-element subsets of $S$. Furthermore, suppose that
(a) each pair of distinct elements of $S$ belongs to at least one triple in $T$,
(b) $|T| \leq \dfrac{v(v-1)}{6}$.
Show that $(S, T)$ is a Steiner triple system.
Assume the contrary and make a list $L$ as follows: For every pair write down the triple with which it is associated. Then $|L|>\tbinom{v}{2}$ as there exists a pair with two triples. Now since each triple is counted by exactly three pairs so $|T|=|L|/3>\frac{\tbinom{v}{2}}{3}$, a contradiction.