Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 36 / 3 / a / Solution

Each unordered triple of vertices is a triangle in a graph with probability $p^3$. Consequently the triangle count in a binomial random graph satisfies $\mathbb{E}X=\left(\genfrac{}{}{0ex}{}{n}{3}\right)p^3\le (np{)}^3/6$. If $np\to 0$, the Markov inequality gives $\mathbb{P}(X>0)\le \mathbb{E}X\to 0$. Hencethe sparse direction of the triangle-existence threshold in a binomial random graph.