 An $(r_0,\ldots,r_k;n)$ 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 $0\leq i\leq k$. The set of all such colorings is denoted ${{\cal R}}(r_0,\ldots,r_k;n)$. The Ramsey number $R(r_0,\ldots,r_k)$ is the least $n>0$ such that no $(r_0,\ldots,r_k;n)$ coloring exists. In 