Improving $R(3,k)$ in just two bites

TL;DR

This paper introduces a new random construction method to generate larger $H$-free graphs with pseudorandom properties, leading to improved lower bounds on off-diagonal Ramsey numbers, approaching conjectured optimal constants.

Contribution

The authors develop a flexible random construction that surpasses the $H$-free process in edge density while maintaining pseudorandomness, improving lower bounds on $R(3,k)$.

Findings

Established $R(3,k) geq (rac{1}{2}+o(1)) rac{k^2}{ ext{log}k}$

Compared new bounds with previous $R(3,k) geq (rac{1}{3}+o(1)) rac{k^2}{ ext{log}k}$

Provided evidence supporting the conjecture that the constant $rac{1}{2}$ is asymptotically tight.

Abstract

We present a flexible random construction which, for certain graphs $H$, is able to produce $H$-free graphs with edge density strictly larger than that of the $H$-free process, while simultaneously preserving pseudorandom properties and allowing a much easier analysis. As our main application, we use this construction to show that the off-diagonal Ramsey numbers satisfy $R(3,k)\ge \left(\frac12+o(1)\right)\frac{k^2}{\log{k}}$, improving the previously best bound $R(3,k)\ge \left(\frac13+o(1)\right)\frac{k^2}{\log{k}}$. While the best known upper bound is $R(3,k)\le \left(1+o(1)\right)\frac{k^2}{\log{k}}$, the constant of $\frac12$ has been conjectured to be asymptotically tight by multiple groups.