AbstractIn a recent paper, Oliver Riordan shows that for and p up to and slightly larger than the threshold for a Kr‐factor, the hypergraph formed by the copies of Kr in G(n, p) contains a copy of the binomial random hypergraph with . 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 . With Riordan's result, this immediately implies an asymptotically sharp bound for the threshold of a Kr‐factor in G(n, p) for . 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).