Many classic coloring problems exhibit the same property.

\[lemma:closed\] Let $(G,{\kappa_1})$ and $(H,{\kappa_2})$ be graph colorings in $k$ colors such that $(G,\kappa_1) \approx_{\pi,\sigma}
  (H,\kappa_2)$. Then, $$(G,\kappa_1) \in {{\cal R}}(r_1,r_2,\ldots,r_k;n) \iff (H,\kappa_2) \in
  {{\cal R}}(\sigma(r_1),\sigma(r_2),\ldots,\sigma(r_k);n)$$

We make use of the following theorem from [@PR98].

\[thm:433\] $30\leq R(4,3,3)\leq 31$ and, $R(4,3,3)=31$ if and only if there exists a $(4,3,3;30)$ coloring $\kappa$ of $K_{30}$ such that: (1) For every vertex $v$ and $i\in\{2,3\}$, $5\leq deg_{i}(v)\leq
  8$, and $13\leq deg_{1}(v)\leq 16$. (2) Every edge in the third color has at least one endpoint $v$ with $deg_{3}(v)=13$. (3) There are at least 25 vertices $v$ for which $deg_{1}(v)=13$, $deg_{2}(v)=deg_{3}(v)=8$.

\[cor:degrees\] Let $G=(V,E)$ be a $(4,3,3;30)$ coloring, $v\in V$ a selected vertex, and assume without loss of generality that $deg_2(v)\geq
  deg_3(v)$. Then, $deg(v)\in{\left\{
      \begin{array}{l}{\langle 13, 8, 8 \rangle},{\langle 14, 8,
      7 \rangle},{\langle 15, 7, 7 \rangle},{\langle 15, 8, 6 \rangle},{\langle 16, 7, 6 \rangle},{\langle 16,
      8, 5 \rangle}\end{array}
      \right\}}$.

Consider a vertex $v$ in a $(4,3,3;n)$ coloring and focus on the three subgraphs induced by the neighbors of $v$ in each of the three colors. The following states that these must be corresponding Ramsey colorings.

\[obs:embed\] Let $G$ be a $(4,3,3;n)$ coloring and $v$ be any 