Uniform non-${\ell }_1^n$-ness of ${\ell }_1$-sums of Banach spaces

We shall characterize the uniform non-${\ell }_1^n$-ness of the ${\ell }_1$-sum $(X_1\oplus \cdots \oplus X_m{)}_1$ of a finite number of Banach spaces $X_1,\dots ,X_m$. Also we shall obtain that $(X_1\oplus \cdots \oplus X_m{)}_1$ is uniformly non-${\ell }_1^{m+1}$ if and only if all $X_1,\dots ,X_m$ are uniformly non-square (note that $(X_1\oplus \cdots \oplus X_m{)}_1$ is not uniformly non-${\ell }_1^m$). Several related results will be presented.