The Maurey-Pisier theorem states that if $p\_X$ is the supremum of those $p$ such that the Banach space $X$ has Rademacher type $p$, then $\ell\_{p\_X}$ is finitely representable in $X$. 


For $1\leq p<\infty$, let us say the Schauder basis $(e\_i)\_{i=1}^\infty$ has \emph{block type} $p$ if there exists a constant $C$ such that for any natural number $n$, any $0=k\_0<\ldots <k\_n$, any scalars $(a\_i)\_{i=1}^{k\_n}$, if $y\_i=\sum\_{j=k\_{i-1}+1}^{k\_i} a\_je\_j$, \begin{equation}\bigl(\int\_0^1 \|\sum\_{i=1}^n r\_i(t)y\_i\|^pdt\bigr)^{1/p} \leq C \bigl(\sum\_{i=1}^n \|y\_i\|^p\bigr)^{1/p}.\end{equation} Here, $(r\_i)\_{i=1}^\infty$ is the sequence of Rademacher functions on $[0,1]$. 


This definition was given on page 24 of the article "Infinite dimensional geometric moduli and type-cotype theory" by V.D. Milman and A Perelson, in the book "Geometric Aspects of Banach Spaces: Essays in Honour of Antonio Plans." 


My question, which doesn't seem to be directly stated in that article, is does the block version of the Maurey-Pisier theorem hold: 


$Q:$ Is it true that, if $p\_X$ is the supremum of those $p\in [1,\infty)$ such that the basis $(e\_i)$ has block type $p$, $\ell\_{p\_X}$ is block finitely representable in $(e\_i)$? 


The article of Milman and Perelson references "the variant of Maurey-Pisier's theorem for blocks of a given sequence as it was done, for example, in [MSch2]." However, in the bibliography of that article, there is no [MSch2] entry. I have looked through some of the bibliography items listed in that article, but I am unable to find anywhere a definitive answer to the question $Q$. 



Block version of Maurey Pisier theorem
I believe that the answer is NO. I do not know a counterexample, however, in positive direction one can do the following.


If the basis is unconditional you can define type/cotype on disjointly supported vectors. Then the corresponding result is true. This is written [here](http://www.math.unt.edu/~bunyamin/pdf/envelopes.pdf) (with slightly different language), see Theorem 5.6.


The reference that you are looking for must be the paper by [Milman and Sharir](http://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=TI&pg6=PC&pg7=ALLF&pg8=ET&review_format=html&s4=milman%20and%20sharir&s5=&s6=&s7=&s8=All&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq&r=2&mx-pid=538129). A warning though, what they call block version is actually disjointly supported version (not successive blocks) i mentioned above.


