Computing the Ramsey number R(4,3,3) using abstraction and symmetry breaking

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 ?(3,3,3;13)$\mathcal {R}(3,3,3;13)$ consisting of 78,892 Ramsey colorings.