  In the present paper we identify those filtered probability spaces
$(\Omega,\, \mathcal{F},\, \left(\mathcal{F}_n\right),\, \mathbb{P})$ that
determine already the martingale type of a Banach space $X$. We isolate
intrinsic conditions on the filtration $(\mathcal{F}_n)$ of purely atomic
$\sigma$-algebras which determine that the upper $\ell^p$ estimates
  \[
  \|f\|_{L^p(\Omega,\, X)}^p\leq C^p\left( \|\mathbb{E}
f|\mathcal{F}_0\|^p_{L^p(\Omega,\, X)}+\sum_{n=1}^{\infty} \|\Delta_n
f\|^p_{L^p(\Omega,\, X)}\right),\qquad f\in L^p(\Omega,X)\] imply that the
Banach space $X$ is of martingale type $p$. Our paper complements \mbox{G.
Pisier's} investigation \cite{Pisier1975} and continues the work by S. Geiss
and second named author in \cite{Geiss2008}.
