# Monochromatic triangle - graph coloring

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!

• Using such methods stops giving tight bounds quickly. This is confirmed by the partial results on various Ramsey numbers. Jan 6, 2019 at 19:22
• my answer at: math.stackexchange.com/questions/935143/… shows all $K_6$ actually have 2 monchromatic triangles in 2 colors, only 3 colorings of higher $K_n$ that force both to break, need be considered.
– user645636
Mar 7, 2019 at 19:45