  In a recent paper, Oliver Riordan shows that for $r \ge 4$ and $p$ up to and
slightly larger than the threshold for a $K_r$-factor, the hypergraph formed by
the copies of $K_r$ in $G(n,p)$ contains a copy of the binomial random
hypergraph $H=H_r(n,\pi)$ with $\pi \sim p^{r \choose 2}$. For $r=3$, he gives
a slightly weaker result where the density in the random hypergraph is reduced
by a constant factor. Recently, Jeff Kahn announced an asymptotically sharp
bound for the threshold in Shamir's hypergraph matching problem for all $r \ge
3$. With Riordan's result, this immediately implies an asymptotically sharp
bound for the threshold of a $K_r$-factor in $G(n,p)$ for $r \ge 4$. In this
note, we resolve the missing case $r=3$ by modifying Riordan's argument. This
means that Kahn's result also implies a sharp bound for triangle factors in
$G(n,p)$.
