Proving that there is always a winner in the game of sim playing with 6 dots.

Question

Suppose each player takes turns drawing a line of different color between two edges(total of 15 edges). The winner is the player who forces the other player to draw a monochromatic triangle. How can you prove that this game can never be tied?

Answer 1

A proof that a two colouring of the edges of the complete graph on six vertices will contain a monochromatic triangle can be found here https://en.wikipedia.org/wiki/Ramsey%27s_theorem#Example:_R.283.2C_3.29_.3D_6