 Solution

Choose a vertex $v$ of $K_6$. At least three of its five incident edges have one colour, say blue, by the pigeonhole principle. Let their other endpoints be $x,y,z$. If one of $xy,yz,zx$ is blue, it completes a blue triangle with $v$; if none is blue, $xyz$ is a yellow triangle. Thus every colouring of $K_6$ has a monochromatic triangle.

The statement fails for $K_5$. Colour the edges of a five-cycle blue and its complementary five-cycle yellow. Neither colour contains a triangle.