Graphs with Many Copies of a Given Subgraph

Let $c>0$, and H be a fixed graph of order r. Every graph on n vertices containing at least cnʳ copies of H contains a "blow-up" of H with $r-1$ vertex classes of size c^r²ln n and one vertex class of size greater than n^1-cʳ⁻¹. A similar result holds for induced copies of H.