No Dense Subgraphs Appear in the Triangle-free Graph Process

Consider the triangle-free graph process, which starts from the empty graph on n vertices and in every step an edge is added that is chosen uniformly at random from all non-edges that do not form a triangle with the existing edges. We will show that there exists a constant c such that asymptotically almost surely no copy of any fixed finite triangle-free graph on k vertices with at least ck edges appears in the triangle-free graph process.