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 K t , t , the complete bipartite graph of size 2t. Since K t , t is a t-blowup of an edge, the following intriguing open question arises: Is it true that of all graphs with triangle-density p 3 , the random graph G ( n , p ) contains close to the smallest density of K t , t , t , which is the t-blowup of a triangle? in 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 p 3 , then there is some 2 ⩽ t ⩽ T ( p ) for which the density of K t , t , t in G is at least p ( 3 + o ( 1 ) ) t 2 , which (up to the o ( 1 ) term) equals the density of K t , t , t in G ( n , p ) . We also raise several open problems related to these problems and discuss some applications to other areas.