I'm trying to find the smallest $n_c$ for which the problem of proving a complete graph with $n$ vertices with edges colored with $c$ colors has a monochromatic triangle could be simplified to a problem of proving the same for a smaller graph of $m_{c-1} < n$ vertices and $c-1$ colors using the Pigeonhole principle. $n$ should be a function of $c$.
This is similar to how the problem of proving that in a graph with 17 vertices where the edges are colored with 3 colors we could find a triangle can be simplified to proving that in a complete graph with 6 vertices and 2 colors there is a monochromatic triangle. Appreciate any help you could provide!