  Let $I$ be an ideal of a polynomial algebra $S$ over a field generated by
square free monomials of degree $\geq d$. If $I$ contains more monomials of
degree $d$ than $(n-d)/(n-d+1)$ of the total number of square free monomials of
$S$ of degree $d+1$ then $\depth_SI\leq d$, in particular the Stanley's
Conjecture holds in this case.
