A sharp threshold for the Hamilton cycle Maker–Breaker game

We study the Hamilton cycle Maker–Breaker game, played on the edges of the random graph G ( n , p ). We prove a conjecture from (Stojaković and Szabó, Random Struct and Algorithms 26 (2005), 204–223.), asserting that the property that Maker is able to win this game, has a sharp threshold at log n n . Our theorem can be considered a game‐theoretic strengthening of classical results from the theory of random graphs: not only does G ( n , p ) almost surely admit a Hamilton cycle for p = ( 1 + ε) log n n , but Maker is able to build one while playing against an adversary.