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

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.

💡 Research Summary

The paper “Improving R(3,k) in just two bites” introduces a novel random construction that yields triangle‑free graphs with edge density exceeding that attainable by the classic triangle‑free process, while preserving the pseudorandom properties essential for bounding independence numbers. The authors’ main achievement is a new lower bound for the off‑diagonal Ramsey number:
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