and $G^3_u$ are highlighted by the boldface text in Figure \[embed\_12\_8\_8\].

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 number $R(r_1,\ldots,r_k)$ is the least $n>0$ such that no $(r_1,\ldots,r_k;n)$ coloring exists. In the multicolor case ($k>2$), the only known value of a nontrivial Ramsey number is $R(3,3,3)=17$. Prior to this paper, it was known that $30\leq R(4,3,3)\leq 31$. Moreover, while the sets of $(3,3,3;n)$ colorings were known for $14\leq n\leq 16$, the set of colorings for $n=13$ was never published.[^2] More information on recent results concerning Ramsey numbers can be found in the electronic dynamic survey by Radziszowski [@Rad].

$$\begin{aligned}
      \varphi_{adj}^{n,k}(A) &=& \hspace{-2mm}\bigwedge_{1\leq q<r\leq n}
                          \left(\begin{array}{l}
                             1\leq A_{q,r}\leq k  ~~\land~~                             A_{q,r} = A_{r,q} ~~\land ~~                             A_{q,q} = 0
                           \end{array}\right)
     \label{constraint:simple}
\\
     \varphi_{r}^{n,c}(A) &=& 
                \bigwedge_{I\in \wp_r([n])}
                                \bigvee {\left\{~A_{i,j}\neq c  \left|
      \begin{array}{l}i,j \in I, i<j\end{array}
    \right.     \right\}}
     \label{constraint:nok}\end{aligned}$$ $$\begin{aligned}
\small
\label{constraint:coloring}
    \varphi_{(r_1,\ldots,r_k;n)}(A) & = & \varphi_{adj}^{n,k}(A) \land  \hspace{-2mm}
                      \bigwedge_{1\leq c\leq k} \hspace{-1mm}
                      \varphi_{r_c}^{n,c}(A)\end{aligned}$$

A graph coloring problem on $k$ colors is about the search for 