On the clique-game

We study Maker/Breaker games on the edges of the complete graph, as introduced by Chvátal and Erdős. We show that in the ( m : b ) game played on K N , the complete graph on N vertices, Maker can achieve a K q for q = ( m log 2 ( b + 1 ) − o ( 1 ) ) ⋅ log 2 N , which partially solves an open problem by Beck. Moreover, we show that in the ( 1 : 1 ) game played on K N for a sufficiently large N , Maker can achieve a K q in only 2 2 q 3 poly ( q ) moves, which improves the previous best bound and answers a question of Beck. Finally, we consider the so called tournament game. A tournament is a directed graph where every pair of vertices is connected by a single directed edge. The tournament game is played on K N . At the beginning, Breaker fixes an arbitrary tournament T q on q vertices. Maker and Breaker then alternately take turns in claiming an unclaimed edge e and selecting one of the two possible orientations. Maker wins if his graph contains a copy of the goal tournament T q ; otherwise Breaker wins. We show that Maker wins the tournament game on K N with q = ( 1 − o ( 1 ) ) log 2 N . This supports the random graph intuition, which suggests that the threshold for q is asymptotically the same for the game played by two “clever” players and the game played by two “random” players.