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arXiv · 2510.19718

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

Zion Hefty·Paul Horn·Dylan King·Florian Pfender
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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.

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Zion Hefty, Paul Horn, Dylan King, Florian Pfender. 2025-10-22. Improving $R(3,k)$ in just two bites. https://arxiv.org/abs/2510.19718

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