Which of the players will first draw a triangle? 6 vertices are given. No edges are given at first. Two players play the following game: the first player draws one black edge. Then the second player draws one green edge. Then the first player draws one black edge again, and so on. The game ends when one of the players has drawn a triangle with all edges the same color. The winner is his opponent. Which of the players always wins using the best strategy? What is his winning strategy?
Maybe I can use some results from graph theory for finding the winning strategy, but I cannot see anything useful.
 A: The first player loses. 
This is a particular instances of a game called Sim, corresponding to the fact that the Ramsey number $R(3,3)$ is $6$, it was first introduced by Gustavus Simmons in 1969.
For much more, including variants, classification of some of these variants in terms of computational complexity, and many open questions, see 

Wolfgang Slany. Graph Ramsey games. ArXiv:cs/9911004.

A pictorial argument describing a run of the game that ends in win for the second player in Sim for 6 vertices is presented in page 3 of the linked preprint. Theorem 22 (in page 32) is the assertion that this holds in general. The proof is described as "more or less a very long enumeration of cases" (analyzing 2309 positions). The result seems to have been first established in

Ernest Mead, Alexander Rosa, and Charlotte Huang. The game of SIM: A winning strategy for the second player, Mathematics Magazine, 47 (5), (1974), 243–247.

No strategy other than very long cases analysis seems currently known.
