  A homogeneous ideal $I$ of a polynomial ring $S$ is said to have the Rees
property if, for any homogeneous ideal $J \subset S $ which contains $I$, the
number of generators of $J$ is smaller than or equal to that of $I$. A
homogeneous ideal $I \subset S$ is said to be $\mathfrak m$-full if $\mathfrak
mI:y=I$ for some $y \in \mathfrak m$, where $\mathfrak m$ is the graded maximal
ideal of $S$. It was proved by one of the authors that $\mathfrak m$-full
ideals have the Rees property and that the converse holds in a polynomial ring
with two variables. In this note, we give examples of ideals which have the
Rees property but are not $\mathfrak m$-full in a polynomial ring with more
than two variables. To prove this result, we also show that every Artinian
monomial almost complete intersection in three variables has the Sperner
property.
