An upper bound on the extremal version of Hajnals triangle-free game

A game starts with the empty graph on $n$ vertices, and two player alternate adding edges to the graph. Only moves which do not create a triangle are valid. The game ends when a maximal triangle-free graph is reached. The goal of one player is to end the game as soon as possible, while the other player is trying to prolong the game. With optimal play, the length of the game (number of edges played) is called the $K_3$ game saturation number. In this paper we prove an upper bound for this number.