  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.
