  In this article, we show constructively that Morrey spaces are not uniformly
non-$\ell^1_n$ for any $n\ge 2$. This result is sharper than those previously
obtained in \cite{GKSS, MG}, which show that Morrey spaces are not uniformly
non-square and also not uniformly non-octahedral. We also discuss the $n$-th
James constant $C_{{\rm J}}^{(n)}(X)$ and the $n$-th Von Neumann-Jordan
constant $C_{{\rm NJ}}^{(n)}(X)$ for a Banach space $X$, and obtain that both
constants for any Morrey space $\mathcal{M}^p_q(\mathbb{R}^d)$ with $1\le
p<q<\infty$ are equal to $n$.
