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 $1<p<\infty.$ 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.



Banach space containing uniformly complementend $\ell_p^n$s
You can find many relevant results in the book: Pisier, Gilles
Factorization of linear operators and geometry of Banach spaces.
CBMS Regional Conference Series in Mathematics, 60. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1986. See, for example, Theorem 10.6 and Corollary 10.8.


The mentioned book can be used as a good starting point for learning numerous known results related to your question.


