On unbiased games on random graphs

We study unbiased Maker-Breaker positional games played on the edges of the random graph G(n, p). As the main result of the paper, we prove a conjecture from [18], that the property that Maker is able to win the Hamiltonicity game played on a random graph G(n, p) 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 G(n, p) almost surely contains a Hamilton cycle for p = (1 + ε) log n n , but Maker is able to build one while playing against an adversary. We also consider the H-game, in which Maker’s goal is to build a graph that contains a fixed graph H as a subgraph. We show that Maker almost surely wins the H-game on G(n, p), provided pC ≥ n−m2(H) , for a sufficiently large constant C > 0. We prove that this result is essentially tight for graphs H whose degeneracy number is at least m2(H) + 12 . Here m2(H) = max H′⊆H v(H′)≥3 e(H′)−1 v(H′)−2 is the 2-density of H.