# Banach space containing uniformly complementend $\ell_p^n$s

Let $$X$$ be a Banach space such that both $$X$$ and $$X^*$$ have finite cotype. Also assume that $$X$$ is an inductive limit of finite dimensional Banach spaces $$X_n\subseteq X_{n+1}.$$ Fix $$1 Is there any known result which can give precise information about finite dimensional subspaces $$Y_n$$'s of $$X$$ with $$\sup\limits_n\text{dim}Y_n=\infty$$, $$Y_n$$'s have Banach-Mazur distance $$\lambda_n$$ from $$\ell_p^n$$'s and the projection constants from $$X$$ to $$Y_n$$ have some bound, say $$P_n.$$ I want best possible choice of $$Y_n$$'s so that the bounds of $$\lambda_n$$ and $$P_n$$ in terms of $$n$$ and $$p$$ and best possible.