Van der waerden and ramsey type games

Let us consider the following 2-player game, calledvan der Waerden game. The players alternately pick previously unpicked integers of the interval {1, 2, ...,N}. The first player wins if he has selected all members of ann-term arithmetic progression. LetW*(n) be the least integerN so that the first player has a winning strategy. By theRamsey game on k-tuples we shall mean a 2-player game where the players alternately pick previously unpicked elements of the completek-uniform hypergraph ofN verticesK _N ^k , and the first player wins if he has selected allk-tuples of ann-set. LetR _k*(n) be the least integerN so that the first player has a winning strategy. We prove (W* (n))¹/n → 2,R ₂*(n)<(2+ε) ⁿ andR _k ^* n<2 ^nk / ^k! fork ≧3.