 Binomial random hypergraph

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.