The defining ideal $I_{\bb X}$ of a set of points $\bb{X}$ in $\bb{P}^n_1 x \ldots x \bb{P}n_k$ is investigated with a special emphasis on the case when $\bb{X}$ is in generic position, that is, $\bb{X}$ has the maximal Hilbert function. When $\bb{X}$ is in generic position, the degrees of the generators of the associated ideal $I_{\bb X}$ are determined. $\nu(I_{\bb X})$ denotes the minimal number of generators of $I_{\bb X}$, and this description of the degrees is used to construct a function $v(s;n_1,\ldots,n_k)$ with the property that $\nu(I_{\bb X})\,{\geq}\, v(s;n_1,\ldots,n_k)$ always holds for $s$ points in generic position in $\bb{P}n_1 x \ldots x \bb{P}^n_k$. When $k\,{=}\,1$, $v(s;n_1)$ equals the expected value for $\nu(I_{\bb X})$ as predicted by the ideal generation conjecture. If $k\,{\geq}\, 2$, it is shown that there are cases with $\nu(I_{\bb X}) > v(s;n_1,\ldots,n_k$). However, computational evidence suggests that in many cases $\nu(I_{\bb X})\,{=}\, v(s;n_1,\ldots,n_k$).