An improved method for recursively computing upper bounds for two-colour Ramsey numbers

TL;DR

This paper introduces a recursive method to compute upper bounds for two-colour Ramsey numbers, potentially improving existing bounds and advancing understanding of these combinatorial quantities.

Contribution

The paper presents a novel recursive approach for calculating upper bounds of two-colour Ramsey numbers, enhancing previous methods and providing a framework for future improvements.

Findings

Method can improve known upper bounds

Potential for refining Ramsey number estimates

Framework adaptable to future research

Abstract

The two-colour Ramsey number $R(m,n)$ is the least natural number $p$ such that any graph of order $p$ must contain either a clique of size $m$ or an independent set of size $n$. We exhibit a method for computing upper bounds for $R(m,n)$ recursively, using known upper bounds of $R(\cdot,\cdot)$ with lower values for at least one of the arguments. We also give an example of how this method could be used to improve several of the best known bounds that are available in the literature (which however soon will be obsolete due to a forthcoming work).