Dynamic concentration of the triangle-free process

TL;DR

This paper analyzes the triangle-free process, determining the maximum edges, bounding the independence number, and identifying subgraphs likely to appear, leading to improved bounds on Ramsey numbers.

Contribution

It provides the asymptotic number of edges in the maximal triangle-free graph and bounds on the independence number, improving previous results and analyzing subgraph appearance.

Findings

Maximum edges in the process are asymptotically determined

Bound on the independence number improves lower bounds on R(3,t)

Identifies subgraphs with density at most 2 as likely to appear

Abstract

The triangle-free process begins with an empty graph on n vertices and iteratively adds edges chosen uniformly at random subject to the constraint that no triangle is formed. We determine the asymptotic number of edges in the maximal triangle-free graph at which the triangle-free process terminates. We also bound the independence number of this graph, which gives an improved lower bound on the Ramsey numbers R(3,t): we show R(3,t) > (1-o(1)) t^2 / (4 log t), which is within a 4+o(1) factor of the best known upper bound. Our improvement on previous analyses of this process exploits the self-correcting nature of key statistics of the process. Furthermore, we determine which bounded size subgraphs are likely to appear in the maximal triangle-free graph produced by the triangle-free process: they are precisely those triangle-free graphs with density at most 2.