  We study the arithmetically Cohen-Macaulay (ACM) property for finite sets of
points in multiprojective spaces, especially $(\mathbb P^1)^n$. A combinatorial
characterization, the $(\star)$-property, is known in $\mathbb P^1 \times
\mathbb P^1$. We propose a combinatorial property, $(\star_n)$, that directly
generalizes the $(\star)$-property to $(\mathbb P^1)^n$ for larger $n$. We show
that $X$ is ACM if and only if it satisfies the $(\star_n)$-property. The main
tool for several of our results is an extension to the multiprojective setting
of certain liaison methods in projective space.
