Question: Let $H(n,p)$ be the random 3-uniform hypergraph on $n$ vertices where every hyperedge (of which there are $\binom{n}{3}$) is present with probability $p$. What is the approximate order of the largest $k=k(n)$ such that $H(n,1/2)$ will usually contain a k-clique?
Hint: You only need the first moment method and should estimate $k$ up to a constant factor
Attempts so far: If $X_n$ is the number of $k$-cliques in $H(n,1/2)$, then $\mathbb{E}X_n = \binom{n}{k} p^\binom{k}{3}\sim n^k/ (2^{k^3}k!)$. Presumably when it says 'will usually contain' I take that to mean $\mathbb{E}X_n$ doesnt't go to zero, or $\mathbb{E}X_n \to \infty$ or something like that but it doesn't seem clear to me? But I don't see how I can get out a suitable $k$ from the approximation for $\mathbb{E}X_n$? Any help would be appreciated.