A new lower bound for the Ramsey numbers $R(3,k)$

TL;DR

The paper proves a new lower bound for the off-diagonal Ramsey numbers R(3,k) by constructing a dense, triangle-free graph via a seed graph blow-up followed by a refined triangle-free nibble with a regularization step. This seeded construction yields a smaller independence number than the triangle-free process alone, enabling R(3,k) to be bounded below by (1/3+o(1)) k^2/log k and narrowing the gap to the upper bound of Hereditary Shearer. The authors develop a tractable analytic framework that controls degrees, codegrees, and subgraph occurrences through a quasi-random nibble, plus derandomization and martingale concentration to bound the probability that a k-set remains independent. They disprove the conjecture that the 1/4-constant is sharp and provide a path toward further tightening the Ramsey bound, with implications for the optimal balance of randomness and structure in triangle-free constructions.

Abstract

We prove a new lower bound for the off-diagonal Ramsey numbers, \[ R(3,k) \geq \bigg( \frac{1}{3}+ o(1) \bigg) \frac{k^2}{\log k }\, , \] thereby narrowing the gap between the upper and lower bounds to a factor of $3+o(1)$. This improves the best known lower bound of $(1/4+o(1))k^2/\log k$ due, independently, to Bohman and Keevash, and Fiz Pontiveros, Griffiths and Morris, resulting from their celebrated analysis of the triangle-free process. As a consequence, we disprove a conjecture of Fiz Pontiveros, Griffiths and Morris that the constant $1/4$ is sharp.

Theorems & Definitions (76)

• Theorem 1.1
• Conjecture 1.2
• Theorem 3.1
• Lemma 4.1
• proof
• proof
• proof
• proof
• proof
• Lemma 5.1