Let $K_9$ be the complete $9$ vertex graph. Each edge can be coloured red, blue or left uncoloured. Find the smallest $n$ such that when $n$ edges are coloured, there necessarily must be a monochromatic triangle.
After going optimal algorithm, it seems $30$ seems to be the upper bound, and $31$ or more edges will always produce a monochromatic triangle. I'm not sure how to prove that is the case, but I suspect pigeonhole is used. Can anyone help me prove that $31$ or more edges guarantees a monochromatic triangle?