---
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 *abstraction* and *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.'
author:
- Michael Codish
- Michael Frank
- Avraham Itzhakov
- Alice Miller
title: 'Computing the Ramsey Number R(4,3,3) using Abstraction and Symmetry breaking[^1]'
---

Introduction {#sec:intro}
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This paper introduces a general methodology that applies to solve graph edge-coloring problems and demonstrates its application in the search for Ramsey numbers. These are notoriously hard graph coloring problems that involve assigning colors to the edges of a complete graph. An $(r_1,\ldots,r_k;n)$ Ramsey coloring is a graph coloring in $k$ colors of the complete graph $K_n$ that does not contain a monochromatic complete sub-graph $K_{r_i}$ in color $i$ for each $1\leq i\leq k$. The set of all such colorings is denoted ${{\cal R}}(r_1,\ldots,r_k;n)$. The Ramsey 