Ideal Containments under Flat Extensions

Let $\phi :S=k[y_0,\mathrm{.}\mathrm{.}\mathrm{.},y_n]\to R=k[y_0,\mathrm{.}\mathrm{.}\mathrm{.},y_n]$ be given by $y_i \to f_i$where$f_0,…,f_n$isan$R$-regular sequence of homogeneous elements of the same degree. A recent paper shows for ideals, $I_{\mathrm{\Delta }}\subseteq S$, of matroids, $\mathrm{\Delta }$, that $I_{\mathrm{\Delta }}^{(m)}\subseteq I^r$ if and only if ${\phi }_{\ast }(I_{\mathrm{\Delta }}{)}^{(m)}\subseteq {\phi }_{\ast }(I_{\mathrm{\Delta }}{)}^r$ where ${\phi }_{\ast }(I_{\mathrm{\Delta }})$ is the ideal generated in $R$ by $\phi (I_{\mathrm{\Delta }})$. We prove this result for saturated homogeneous ideals $I$ of configurations of points in ${\mathbb{P}}^n$ and use it to obtain many new counterexamples to $I^{(rn-n+1)}\subseteq I^r$ from previously known counterexamples.