Let $\ell_p$ denote the usual infinite-dimensional sequence space, and if $n\in\mathbb{Z}^+$ then we let $\ell_p^n$ denote its $n$-dimensional counterpart.
Conjecture. Let $1\leq p<\infty$. There exists a positive real constant $C_p\in[1,\infty)$, depending only on $p$, such that for every positive integer $n\in\mathbb{Z}^+$ there is another positiver integer $k(n)\in\mathbb{Z}^+$ with the property that if $F\subseteq\ell_p$ is any $k(n)$-dimensional subspace of $\ell_p$ then there is an embedding (a linear map invertible on its range) $U:\ell_p^n\to F$ with $\|U\|\|U^{-1}\|\leq C_p$ (in other words, $F$ contains an $n$-dimensional subspace with Banach-Mazur distance $\leq C_p$ from $\ell_p^n$).
This is almost certainly known already, in which case I seek a reference. Dvoretzky's Theorem gives an affirmative answer when $p=2$. I am particularly interested in the case $p=1$.
Thanks!
This doesn't sound right to me. Note that for $p\in [1,2)$ the space $\ell_p$ contains almost isometric copies of $\ell_2^n$ for all $n$. Indeed, take the linear span of a sequence of zero-mean and zero-variance Gaussians in $L_p$ which is isometric to $\ell_2$ and use that that $L_p$ is finitely-representable in $\ell_p$.
Thus, taking for $F$ almost isometric copies of $\ell_2^n$ you cannot expect to find $\ell_p^{k(n)}$ in it unless $p=2$.
(Of course, as Theo pointed out, this extends to all Banach spaces by the Dvoretzky theorem.)