  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,
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, and focus on those cases where $H$ is a clique or a cycle. 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.
