Abstract. By a Ramsey-type game is meant a game in which two players (the constructor and the destroyer) alternatively pick previously unpicked edges of the complete graph on n vertices, and the constructor wins if and only if he has selected all edges of a prescribed k-vertex graph G. We prove that the constructor wins if G is an n-vertex path (n>4) or a cycle (n>14), or if G is an n-vertex tree having some special properties.