I am not sure, but if there are no three points on the same line, wouldn't the one who started will lose always, it seems kind of trivial.
How would you win a Ramsey game (e.g. one where you have two players and each has one color). There are six points such that no three lie in a line. The players take turns. The first player to draw a triangle in his own color loses.
Is seems that the first player who plays has the advantage.