The random graph intuition for the tournament game

In the tournament game two players, called Maker and Breaker, alternately take turns in claiming an unclaimed edge of the complete graph Kn and selecting one of the two possible orientations. Before the game starts, Breaker fixes an arbitrary tournament Tk on k vertices. Maker wins if, at the end of the game, her digraph contains a copy of Tk; otherwise Breaker wins. In our main result, we show that Maker has a winning strategy for k = (2-o(1))log2 n, improving the constant factor in previous results of Beck and the second author. This is asymptotically tight since it is known that for k = (2-o(1))log2 n Breaker can prevent the underlying graph of Maker's digraph from containing a k-clique. Moreover, the precise value of our lower bound differs from the upper bound only by an additive constant of 12. We also discuss the question of whether the random graph intuition, which suggests that the threshold for k is asymptotically the same for the game played by two 'clever' players and the game played by two 'random' players, is supported by the tournament game. It will turn out that, while a straightforward application of this intuition fails, a more subtle version of it is still valid. Finally, we consider the orientation game version of the tournament game, where Maker wins the game if the final digraph-also containing the edges directed by Breaker-possesses a copy of Tk. We prove that in that game Breaker has a winning strategy for k = (4 + o(1))log2 n.