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

Year: 2025

Type: preprint

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.

Source: arXiv (Cornell University)

DOI: 10.48550/arxiv.2510.19718

Authors: Hefty, Zion, Horn, Paul, King, Dylan, Pfender, Florian

Institutions: University of Denver, California Institute of Technology, University of Johannesburg, University of Colorado Denver

Topic: Limits and Structures in Graph Theory

Subfield: Discrete Mathematics and Combinatorics

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funders: National Science Foundation, Colorado State University, Iowa State University

Awards: FRG: Collaborative Research: Extremal Combinatorics and Flag Algebras, FRG: Collaborative Research: Extremal Combinatorics and Flag Algebras, Collaborative Research: AF: Small: Sampling and Optimization under Global Constraints

OpenAlex ID: W4416248485

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