  The main object of this paper is to determine the maximum number of $\{0,\pm
1\}$-vectors subject to the following condition. All vectors have length $n$,
exactly $k$ of the coordinates are $+1$ and one is $-1$, $n \geq 2k$. Moreover,
there are no two vectors whose scalar product equals the possible minimum,
$-2$. Thus, this problem may be seen as an extension of the classical Erd\H
os-Ko-Rado theorem. Rather surprisingly there is a phase transition in the
behaviour of the maximum at $n=k^2$. Nevertheless, our solution is complete.
The main tools are from extremal set theory and some of them might be of
independent interest.
