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Let $S=K[X_1,...,X_n]$, $K$ a field, and $I \subset S$ a nontrivial monomial ideal generated by squarefree monomials of degree $\geq d$. Then show that depth$_S I \geq d$.

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The property is not trivial. You can find a proof, using filtrations, here: https://arxiv.org/pdf/0712.2308.pdf (see Proposition 3.1)