Monomial ideals with arbitrarily high tiny powers in any number of variables

Powers of (monomial) ideals is a subject that still calls attraction in various ways. Let $I\subset \mathbb{K}[x_1,\dots ,x_n]$ be a monomial ideal and let $G(I)$ denote the (unique) minimal monomial generating set of $I$. How small can $\mathrm{\mid }G(I^i)\mathrm{\mid }$ be in terms of $\mathrm{\mid }G(I)\mathrm{\mid }$? We expect that the inequality $\mathrm{\mid }G(I^2)\mathrm{\mid }>\mathrm{\mid }G(I)\mathrm{\mid }$ should hold and that $\mathrm{\mid }G(I^i)\mathrm{\mid }$, $i\ge 2$, grows further whenever $\mathrm{\mid }G(I)\mathrm{\mid }\ge 2$. In this paper we will disprove this expectation and show that for any $n$ and $d$ there is an $\mathfrak{m}$-primary monomial ideal $I\subset \mathbb{K}[x_1,\dots ,x_n]$ such that $\mathrm{\mid }G(I)\mathrm{\mid }>\mathrm{\mid }G(I^i)\mathrm{\mid }$ for all $i\le d$.