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We introduce a new type of positional games, played on a vertex set of agraph. Given a graph $G$, two players claim vertices of $G$, where the outcomeof the game is determined by the subgraphs of $G$ induced by the verticesclaimed by each player (or by one of them). We study classical positional gamessuch as Maker-Breaker, Avoider-Enforcer, Waiter-Client and Client-Waiter games,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 targetsets are the vertex sets of all graphs containing a copy of a fixed graph $H$,called $H$-games, and focus on those cases where $H$ is a clique or a cycle. Weshow that, similarly to the edge version of $H$-games, there is a strongconnection between the threshold probability for these games and the one forthe corresponding vertex Ramsey property (that is, the property that every$r$-vertex-coloring of $G(n,p)$ spans a monochromatic copy of $H$). Anothersimilarity to the edge version of these games we demonstrate, is that the gamesin which $H$ is a triangle or a forest present a different behavior compared tothe general case.

$H$-games played on vertex sets of random graphs