Question: Does there exist a function $C:~(0,1)\to (0,\infty)$ such that for each $\varepsilon\in(0,1)$ every Banach space $X$ of dimension $\ge C(\varepsilon)\log n$ contains an $n$-point set $\{x_i\}_{i=1}^{n}$ such that $1-\varepsilon\le ||x_i-x_j||\le 1+\varepsilon$ for $i\ne j$?

The Johnson-Lindenstrauss lemma contains a proof of existence of such function if the class of Banach spaces is restricted to finite dimensional Hilbert spaces. The existence of such function for $\{\ell_p^n\}_{n=1,~1\le p\le\infty}^{\infty}$, was proved using some basic facts of coding theory by Bartal, Linial, Mendel, and Naor [Eur. J. Comb. 25, No. 1, 87-92 (2004; Zbl 1042.54020)]

1more comment