Upper bounds for the number of facets of a simplicial complex

Here we study the maximal dimension of the annihilator ideals
$0:_{A}m^{j}$ of artinian graded rings $A = P / (I, x_1^2, x_2^2, \ldots , x_v^2)$ with a given Hilbert function, where $P$ is the polynomial ring in the variables $x_1, x_2, \ldots , x_v$ over a field $K$ with each $\deg x_i = 1$, $I$ is a graded ideal of $P$, and $m$ is the graded maximal ideal of $A$. As an application to combinatorics, we introduce the notion of $j$-facets and obtain some informations on the number of $j$-facets of simplicial complexes with a given $f$-vector.