A Strategy for Maker in the Clique Game which Helps to Tackle some Open Problems by Beck

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) clique game played on KN , the complete graph on N vertices, Maker can achieve a Kq for q = ( m log 2 (b+1) − o(1) ) · logN , which partially solves an open problem by Beck. Moreover, we show that in the (1:1) clique game played on KN for a sufficiently large N , Maker can achieve a Kq in only O(2 2q 3 ) 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 KN . At the beginning Breaker fixes an arbitrary tournament Tq on q vertices. Maker and Breaker then alternately take turns at claiming one unclaimed edge e and selecting one of the two possible orientations. Maker wins if his graph contains a copy of the goal tournament Tq; otherwise Breaker wins. We show that Maker wins the tournament game on KN with q = (1 − o(1)) log2 N which supports the random graph intuition: the threshold for q is asymptotically the same for the game played by two “clever” players and the game played by two “random” players. This last result solves an open problem of Beck which he included in his list of the seven most humiliating open problems.