Let $R=\mathbb C[x,y]$. Let $\mathfrak m=(x,y)$. Let $I$ be a homogeneous ideal of $R$ with $\sqrt I=\mathfrak m$.
Let $c\ge 1$ be the smallest integer such that $(I \cap \mathfrak m^c)\mathfrak m^{n-c}=I \cap \mathfrak m^n, \forall n \ge c$ . ( $c$ exists by Artin-Rees Lemma )
If $s,t\ge 1$ are integers such that $(I \cap \mathfrak m^t)\mathfrak m^{ts}=\mathfrak m^{ts+t}$ , then is it true that $t \ge c$ ?
If this is not true in general for homogeneous ideals $I$, is it at least true if we also assume $I$ is a monomial ideal ?