  In this paper we study the arithmetically Cohen-Macaulay (ACM) property for
sets of points in multiprojective spaces. Most of what is known is for $\mathbb
P^1\times \mathbb P^1$ and, more recently, in $(\mathbb P^1)^r.$ In $\mathbb
P^1\times \mathbb P^1$ the so called inclusion property characterizes the ACM
property. We extend the definition in any multiprojective space and we prove
that the inclusion property implies the ACM property in $\mathbb P^m\times
\mathbb P^n$. In such an ambient space it is equivalent to the so-called
$(\star)$-property. Moreover, we start an investigation of the ACM property in
$\mathbb P^1\times \mathbb P^n.$ We give a new construction that highlights how
different the behavior of the ACM property is in this setting.

Algebraic Geometry;Commutative Algebra