A threshold for the Maker-Breaker clique game

We look at Maker-Breaker k-clique game played on the edge set of the random graph G(n, p). In this game, two players, Maker and Breaker, alternately claim unclaimed edges of G(n, p), until all the edges are claimed. Maker wins if he claims all the edges of a k-clique; Breaker wins otherwise. We determine that the property that Maker can win this game has a threshold at n− 2 k+1 , for all k > 3. For k = 3 it was previously known that the threshold is n− 5 9 . This is joint work with Tobias Müller.