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

TL;DR

This work advances the understanding of off-diagonal Ramsey numbers by proving a nibble-free lower bound: $R(3,k) \ge \left(\tfrac{1}{2}+o(1)\right)\frac{k^2}{\log k}$. The authors construct a triangle-free graph on $n$ vertices with independence number < $(1+o(1))\sqrt{n\log n}$ using a novel two-layer random blow-up (overlayed two copies of $G(N,p)$ via a co-normal product and a random embedding), avoiding traditional nibble methods. This approach not only yields the improved bound but also extends to hypergraphs, yielding bounds for $R(S_4^{(3)},S_k^{(3)})$ with similar constants, signaling broader applicability of the technique. The results bolster the conjecture that $R(3,k)$ is asymptotically $\tfrac{1}{2}\frac{k^2}{\log k}$ and relate to Shearer’s upper bound and potential algorithmic implications for finding large independent sets in pseudo-random graphs.

Abstract

We present a random construction proving that the extreme off-diagonal Ramsey numbers satisfy $R(3,k)\ge \left(\frac12+o(1)\right)\frac{k^2}{\log{k}}$. This bound has been conjectured to be asymptotically tight, and improves the previously best bound $R(3,k)\ge \left(\frac13+o(1)\right)\frac{k^2}{\log{k}}$. In contrast to all previous constructions achieving the correct order of magnitude, we do not use a nibble argument.

Theorems & Definitions (24)

• Conjecture 1.1: Campos, Jenssen, Michelen and Sahasrabudhe campos2025
• Theorem 1.2
• Theorem 1.3
• Lemma 3.1
• proof : Proof of Lemma \ref{['lem:fiber_and_degree']}
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Lemma 3.4