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Let $\lambda$ be a large enough cardinal number (assuming GCH it suffices tolet $\lambda=\aleph_\omega$). If $X$ is a Banach space with$\text{dens}(X)\ge\lambda$, which admits a coarse (or uniform) embedding intoany $c_0(\Gamma)$, 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 $X$ hasa symmetric basis, we may even conclude that it is linearly isomorphic with$c_0(\text{dens}X)$.

Coarse embeddings into $c_0(\Gamma)$