On the density of a graph and its blowup

It is well known that, of all graphs with edge-density p, the random graph G(n,p) contains the smallest density of copies of Kt,t, the complete bipartite graph of size 2 t. Since Kt,t is a t-blowup of an edge, the following intriguing open question arises: Is it true that of all graphs with triangle-density p3, the random graph G(n,p) contains close to the smallest density of Kt,t,t, which is the t-blowup of a triangle? Our main result gives an indication that the answer to the above question is positive by showing that for some blowup, the answer must be positive. More formally we prove that if G has triangle-density p3, then there is some 2≤t≤T(p) for which the density of Kt,t,t in G is at least p(3+o(1))t2, which (up to the o(1) term) equals the density of Kt,t,t in G(n,p). We also raise several open problems related to these problems and discuss some applications to other areas.