Depth and minimal number of generators of square free monomial ideals

Let $I$ be an ideal of a polynomial algebra $S$ over a field generated by square free monomials of degree $\ge d$. If $I$ contains more monomials of degree $d$ than $(n-d)\mathrm{/}(n-d+1)$ of the total number of square free monomials of $S$ of degree $d+1$ then ${\text{depth}}_{S}I\le d$, in particular the Stanley’s Conjecture holds in this case.