Erd\H os-Ko-Rado theorem for $\{0,\pm 1\}$-vectors

The main object of this paper is to determine the maximum number of ${0,\pm 1}$-vectorssubjecttothefollowingcondition\mathrm{.}Allvectorshavelength$n$, exactly $k$ of the coordinates are $+1$ and one is $-1$, $n\ge 2k$. 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.