On positional games

Let {A_i} be a family of sets and let S = ∩_iA_i. By a positional game we shall mean a game played by two players on {A_i}. The players alternately pick elements of S and that player wins who fist has all the elements of one of the A_i. This paper deals with almost disjoint hypergraphs only, i.e., |A_i∪A_j| ? 1 if i ≠ j. Let $\text{M}{}^1\text{(n)}$ be the smallest integer for which there is an almost disjoint n-uniform hypergraph $\text{|T| = M}{}^1\text{(n)}$, so that the first player has a winning strategy. It is shown that $\text{lim}{}_{\mathrm{n}}\text{M}{}^1\text{(n)]}{}^{\mathrm{<mtext>1</mtext><mtext>n</mtext>}}\text{= 4}$, which was conjectured by Erdös. The same method is applied to prove a conjecture of Hales and Jewett on r-dimensional tick-tack-toe if r is large enough. Finally we prove that for an arbitrary almost disjoint n-uniform hypergraph the second player has such a strategy that the first player unable to win in his mth move if m < (2 ? ?)ⁿ.