H-games played on vertex sets of random graphs

We introduce a new type of positional games, played on a vertex set of a graph. Given a graph G, two players claim vertices of G, where the outcome of the game is determined by the subgraphs of G induced by the vertices claimed by each player (or by one of them). We study classical positional games such as Maker-Breaker, Avoider-Enforcer, Waiter-Client, and Client-Waiter games, in both their biased and unbiased versions, where the board of the game is the vertex set of the binomial random graph G \sim G(n, p). Under these settings, we consider those games where the target sets are the vertex sets of all graphs containing a copy of a fixed graph H, called H-games. We focus on the cases in which H is a clique, a cycle, or a forest, and for Waiter-Client and monotone AvoiderEnforcer games our results apply to any fixed graph H. We show that, similarly to the edge version of H-games, there is a strong connection between the threshold probability for these games and the one for the corresponding vertex Ramsey property (that is, the property that every r-vertex-coloring of G(n, p) spans a monochromatic copy of H). Another similarity to the edge version of these games we demonstrate is that the games in which H is a triangle or a forest present a different behavior compared to the general case.