### abstract

- We describe the eventual behaviour of the Hilbert function of a set of distinct points in P^{n_1} x ... x P^{n_k}. As a consequence of this result, we show that the Hilbert function of a set of points in P^{n_1} x ... x P^{n_k} can be determined by computing the Hilbert function at only a finite number of values. Our result extends the result that the Hilbert function of a set of points in P^n stabilizes at the cardinality of the set of points. Motivated by our result, we introduce the notion of the_border_ of the Hilbert function of a set of points. By using the Gale-Ryser Theorem, a classical result about (0,1)-matrices, we characterize all the possible borders for the Hilbert function of a set of distinct points in P^1 x P^1.