Biased positional games on hypergraphs

Let G(p, q, H) be the game played on a hypergraph H by two players, who alt ernately choose p and q vertices, respectively. The object of the first pla yer is to claim all vertices of a hyperedge of H, while the second player t ries to prevent him from doing so. We give a sufficient, condition for the first player to win G(p,q, H) played on an r-uniform hypergraph H and argue that this condition is close to optimal. Furthermore, we answer a question of Galvin by proving that, the first player has a winning strategy in G(1, q, H) for each 3-uniform hypergraph H with chromatic number large enough.