Four-clique threshold in a binomial 3-uniform hypergraph ($p^{\ast }(n)=n^{-1}$)

A complete three-uniform hypergraph on four vertices has four edges. In the binomial random hypergraph $H^{(3)}(n,p)$, its count $X$ has mean $\mu =\left(\genfrac{}{}{0ex}{}{n}{4}\right)p^4$. If $np\to 0$, the first moment method gives $\mathbb{P}(X>0)\to 0$. Two distinct four-sets have independent clique indicators unless they share three vertices; in that case their combined edge set has seven triples. There are $O(n^5)$ such ordered pairs, so $\mathrm{Var}(X)\le \mu +O(n^5{p}^7)$. Consequently $\mathrm{Var}(X)/{\mu }^2=O((np{)}^{-4}+(n^{3}p{)}^{-1})\to 0$ if $np\to \infty$. The second moment method then gives $\mathbb{P}(X>0)\to 1$. These two implications establish the threshold function.