some steps for solving the problem: 1) its easy to see that if one of the vertices of the graph has at least 6 edges of its edges in one color then we get a contradiction 2)if one has 5 edges with the same color for example yellow (we call the set of vertices that connected to this vertex with yellow edges $S$ then $G-S$ has no edge with yellow color then ...(it comes from the fact that any edge from $G-S$ to $S$ must be yellow(just need some cases to check) 3)other wise if $n\geq13$ then $n$ must be $13$ and every vertex has 4 edges from every color. then double count the number of pairs of edges that they are adjacent together in one vertex and have the same color we must have $\binom{13}{3}=13.3.\binom{4}{2}$ which is obviously incorrect

It can be proved in a way similar to the proof of $R(3,3)=6$.