Random Triangles In Random Graphs

In a recent paper, Oliver Riordan shows that for r >= 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=Hr(n,pi) with pi similar to p(r(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 >= 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 >= 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).