Computing the Ramsey Number R(4,3,3) using Abstraction and Symmetry breaking

TL;DR

This paper introduces a novel methodology combining abstraction and symmetry breaking to solve complex graph coloring problems, successfully computing the long-standing unknown Ramsey number R(4,3,3)=30.

Contribution

It presents a new approach that effectively computes the Ramsey number R(4,3,3) and the set of colorings for a related problem, advancing computational methods in combinatorics.

Findings

Computed R(4,3,3)=30

Determined the set of 78,892 colorings for R(3,3,3;13)

Demonstrated effectiveness of abstraction and symmetry breaking in graph problems

Abstract

The number $R(4,3,3)$ is often presented as the unknown Ramsey number with the best chances of being found "soon". Yet, its precise value has remained unknown for almost 50 years. This paper presents a methodology based on \emph{abstraction} and \emph{symmetry breaking} that applies to solve hard graph edge-coloring problems. The utility of this methodology is demonstrated by using it to compute the value $R(4,3,3)=30$. Along the way it is required to first compute the previously unknown set ${\cal R}(3,3,3;13)$ consisting of 78{,}892 Ramsey colorings.