The triangle-free process and the Ramsey number $R(3,k)$

TL;DR

This paper analyzes the triangle-free process to establish new asymptotic bounds on the off-diagonal Ramsey number R(3,k), improving previous lower bounds and providing detailed properties of the resulting random graphs.

Contribution

It extends the analysis of the triangle-free process to its asymptotic end, deriving precise edge counts, pseudorandom properties, and improved bounds on R(3,k).

Findings

Proves that the number of edges in G_{n,triangle} is asymptotically (1/(2√2)) n^{3/2} √log n.

Establishes pseudorandom properties of G_{n,triangle}.

Provides a lower bound for R(3,k) of approximately (1/4) k^2 / log k, improving previous results.

Abstract

The areas of Ramsey theory and random graphs have been closely linked ever since Erd\H{o}s' famous proof in 1947 that the 'diagonal' Ramsey numbers $R(k)$ grow exponentially in $k$. In the early 1990s, the triangle-free process was introduced as a model which might potentially provide good lower bounds for the 'off-diagonal' Ramsey numbers $R(3,k)$. In this model, edges of $K_n$ are introduced one-by-one at random and added to the graph if they do not create a triangle; the resulting final (random) graph is denoted $G_{n,\triangle}$. In 2009, Bohman succeeded in following this process for a positive fraction of its duration, and thus obtained a second proof of Kim's celebrated result that $R(3,k) = \Theta \big( k^2 / \log k \big)$. In this paper we improve the results of both Bohman and Kim, and follow the triangle-free process all the way to its asymptotic end. In particular, we shall prove that $$e\big( G_{n,\triangle} \big) \,=\, \left( \frac{1}{2\sqrt{2}} + o(1) \right) n^{3/2} \sqrt{\log n },$$ with high probability as $n \to \infty$. We also obtain several pseudorandom properties of $G_{n,\triangle}$, and use them to bound its independence number, which gives as an immediate corollary $$R(3,k) \, \ge \, \left( \frac{1}{4} - o(1) \right) \frac{k^2}{\log k}.$$ This significantly improves Kim's lower bound, and is within a factor of $4 + o(1)$ of the best known upper bound, proved by Shearer over 25 years ago.