Banach embedding of finite dimensional spaces

Question

Recall that: let $0<r<s<2$, then $\ell_r$ uniformly contains a subspace isomorphic to $\ell_s^m$, $m\ge 1$ (see [JS]).

I am wondering whether are any result for the case when $r>s>2$?

[Johnson, William B.; Schechtman, Gideon Embedding $l_p^m$ into $l_1^m$, Acta Math. 149 (1982), 71--85.][JS]

Answer 1

For $2<r<\infty$, if $\ell_s^n$ embeds uniformly into $\ell_r$ for all $n$, then either $s=r$ or $s=2$. This is basically the localization to finite dimensions of the classical dichotomy theorem of Kadec and Pelczynski. The book of Albiac and Kalton is a good source for this.