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

TL;DR

This paper establishes a new lower bound for the off-diagonal Ramsey numbers R(3,k), improving previous bounds and disproving a conjecture about the sharpness of the constant 1/4, thus narrowing the gap between known bounds.

Contribution

The authors prove a stronger lower bound for R(3,k), reducing the gap to the upper bounds and challenging existing conjectures about the constant in the bound.

Findings

New lower bound R(3,k) ≥ (1/3 + o(1)) k^2 / log k

Disproves the conjecture that the constant 1/4 is sharp

Narrows the gap between upper and lower bounds to a factor of 3+o(1)

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.