No Dense Subgraphs Appear in the Triangle-free Graph Process

Abstract

Consider the triangle-free graph process, which starts from the empty graph on $n$ vertices and a random ordering of the possible $\left(\genfrac{}{}{0ex}{}{n}{2}\right)$ edges; the edges are added in this ordering provided the graph remains triangle free. We will show that there exists a constant $c$ such that no copy of any fixed finite triangle-free graph on $k$ vertices with at least $ck$ edges asymptotically almost surely appears in the triangle-free graph process.