Binomial random hypergraph ($H^{(r)}(n,p)$)

A binomial random $r$-uniform hypergraph on $n$ labelled vertices includes each of the $\left(\genfrac{}{}{0ex}{}{n}{r}\right)$ possible hyperedges independently with probability $p$. Thus indicators of different possible edges are independent Bernoulli random variables. A fixed set of $k$ vertices spans a complete $r$-uniform hypergraph with probability $p^{\left(\genfrac{}{}{0ex}{}{k}{r}\right)}$, and the expected number of such complete sets is $\left(\genfrac{}{}{0ex}{}{n}{k}\right)p^{\left(\genfrac{}{}{0ex}{}{k}{r}\right)}$. Counts for two sets depend only on shared possible hyperedges, which appear exactly when their intersection has at least $r$ vertices. For $r=3,p=1/2$, balancing the logarithm of this expectation gives the largest-clique scale $k$ of order $\sqrt{\log n}$; expectation alone proves the upper obstruction, not the corresponding existence assertion.

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.