  Let $\lambda$ be a large enough cardinal number (assuming GCH it suffices to
let $\lambda=\aleph_\omega$). If $X$ is a Banach space with
$\text{dens}(X)\ge\lambda$, which admits a coarse (or uniform) embedding into
any $c_0(\Gamma)$, then $X$ fails to have nontrivial cotype, i.e. $X$ contains
$\ell_\infty^n$ $C$-uniformly for every $C>1$. In the special case when $X$ has
a symmetric basis, we may even conclude that it is linearly isomorphic with
$c_0(\text{dens}X)$.
