The triangle-free process

TL;DR

This paper analyzes a stochastic process that builds large triangle-free graphs by adding edges randomly, providing new bounds on Ramsey numbers R(3,t) and R(4,t) through probabilistic methods.

Contribution

It introduces a detailed analysis of the triangle-free process and extends techniques to the K_4-free process, improving bounds on certain Ramsey numbers.

Findings

Determines the likely number of edges in the final triangle-free graph.

Shows the process likely produces a Ramsey R(3,t) graph.

Provides a new proof for the lower bound on R(3,t) and improves bounds on R(4,t).

Abstract

Consider the following stochastic graph process. We begin with the empty graph on n vertices and add edges one at a time, where each edge is chosen uniformly at random from the collection of potential edges that do not form triangles when added to the graph. The process terminates at a maximal traingle-free graph. Here we analyze the triangle-free process, determining the likely order of magnitude of the number of edges in the final graph. As a corollary we show that the triangle-free process is very likely to produce a Ramsey R(3,t) graph; that is, our analysis of the triangle-free process gives a new proof of the lower bound on R(3,t) previously established by Jeong Han Kim. The techniques introduced extend to the K_4-free process thereby establishing a small improvement in the best known lower bound on the Ramsey number R(4,t).