Study of Exoo's Lower Bound for Ramsey number $R(5,5)$

TL;DR

This paper revisits Exoo's 1989 work on the lower bound of the Ramsey number R(5,5), providing clearer proofs and analyzing graph constructions to improve understanding of monochromatic subgraphs.

Contribution

It offers an efficient verification method for Exoo's claims and introduces new analysis of graph variations related to R(5,5).

Findings

Confirmed R(5,5) ≥ 43 with detailed proofs

Developed concise verification techniques

Analyzed graph colorings with minimal monochromatic K5

Abstract

We review Exoo's 1989 paper, which demonstrates that a lower bound for the Ramsey number $R(5,5)$ is $43$. We provide an efficient way to verify the claims in the paper, adding detailed proofs. In particular, we replace the reference to computer verification by concise arguments. Using our understanding of the insight behind these proofs, we are also able to analyse variations of the graph constructions to obtain, for example, colourings of $K_{43}$ which have very few monochromatic $K_5$.