  We show that given a positive integer $m$, a real number $p\in\left[
2,\infty\right)$ and $1\leq s<p^{\ast}$ the set of non--multiple $\left(
r;s\right)$--summing $m$--linear forms on $\ell_{p}\times\cdots\times \ell_{p}$
contains, except for the null vector, a closed subspace of maximal dimension
whenever $r<\frac{2ms}{s+2m-ms}$. This result is optimal since for
$r\geq\frac{2ms}{s+2m-ms}$ all $m$--linear forms on $\ell_{p}\times
\cdots\times\ell_{p}$ are multiple $\left( r;s\right)$--summing. In particular,
among other results, we generalize a result related to cotype (from 2010) due
to Botelho \textit{et al.}
