Random triangles in random graphs. Issue 4 (26th May 2021) - Heckel, Annika
Abstract: 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 = H r ( n, π ) with π ∼ 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 ).