No Dense Subgraphs Appear in the Triangle-free Graph Process

TL;DR

This paper proves that in the triangle-free graph process, large dense triangle-free subgraphs almost surely do not appear, establishing a threshold for the density of subgraphs that can occur.

Contribution

It demonstrates a threshold phenomenon for the appearance of fixed dense triangle-free subgraphs in the process, a new insight into the structure of such graphs.

Findings

No dense fixed triangle-free subgraphs appear asymptotically almost surely.

Establishes a constant threshold c for the density of subgraphs that can appear.

Provides probabilistic bounds on the structure of the evolving graph.

Abstract

Consider the triangle-free graph process, which starts from the empty graph on $n$ vertices and a random ordering of the possible ${n \choose 2}$ 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.