This must be surely known but I couldn't locate this problem in the literature. It popped out in *a priori* unrelated approximation problem but if true, would help me greatly.


Let $p\in (1,\infty)$.


Informal version: 



> 
> Do the spaces $\ell\_2^k$ sit well-complemented in all sufficiently large finite-dimenional subspaces of $\ell\_p$?
> 
> 
> 


Formally:



> 
> Does there exist $C>0$ and two sequences of integers $(d\_n)\_{n=1}^\infty$ and $(k\_n)\_{n=1}^\infty$ both increasing to $\infty$ such that each $d\_n$-dimensional subspace $X$ of $\ell\_p$ contains a $k\_n$-dimensional subspace that is $C$-isomorphic to $\ell\_2^{k\_n}$ and $C$-complemented in $\ell\_p$?
> 
> 
> 



Well-complemented copies of $\ell_p^n$
If I understand you correctly, much stronger statement holds for any space with nontrivial type, see Theorem 15.10 in the book of Milman and Schechtman on Asymptotic Theory. 


