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We study the Maker-Breaker $H$-game played on the edge set of the randomgraph $G_{n,p}$. In this game two players, Maker and Breaker, alternately claimunclaimed edges of $G_{n,p}$, until all the edges are claimed. Maker wins if heclaims all the edges of a copy of a fixed graph $H$; Breaker wins otherwise. Inthis paper we show that, with the exception of trees and triangles, thethreshold for an $H$-game is given by the threshold of the corresponding Ramseyproperty of $G_{n,p}$ with respect to the graph $H$.

On the threshold for the Maker-Breaker $H$-game