Triangle-existence threshold in a binomial random graph ($p\asymp n^{-1}$)

If $np\to 0$, the triangle count in a binomial random graph has expectation at most $(np{)}^3/6\to 0$, so the Markov inequality shows there are no triangles in a graph with probability tending to one. If $np\to \infty$, the triangle variance in a binomial random graph divided by the squared mean is $O((np{)}^{-3})+O((n^{2}p{)}^{-1})\to 0$. The second moment method then gives at least one triangle in a graph with probability tending to one. At the intermediate scale $np\to c\in (0,\infty )$, a nontrivial Poisson limit for triangles in a binomial random graph describes the count.