This question already has an answer here:
Consider all colorings of the edges of K6 such that every edge is either colored red or blue. Prove or disprove: there always exist at least two monochromatic triangles in any 2-coloring of the edges of K6.
So I have already proved using the pigeonhole principle that K6 must have at least one monochromatic triangle, so now I am wondering if it must also have two. I am currently trying to see if I can draw one without two monochromatic triangles, because that seems like it would be an easy way to disprove it, but that's getting very complicated.
I can't quite figure out where to go from here.