Abstract
Let {
A
i
} be a family of sets and let
S = ∩
i
A
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
M
∗(n)
be the smallest integer for which there is an almost disjoint
n-uniform hypergraph
|T| = M
∗(n)
, so that the first player has a winning strategy. It is shown that
lim
n [M
∗(n)]
1
n
= 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 −
ϵ)
n
.