This is an announcement for the paper “Coarse embeddings into $c_0(\Gamma)$” by Petr Hajekhttps://arxiv.org/find/math/1/au:+Hajek_P/0/1/0/all/0/1, Thomas Schlumprechthttps://arxiv.org/find/math/1/au:+Schlumprecht_T/0/1/0/all/0/1.
Abstract: Let $\lambda$ be a large enough cardinal number (assuming GCH it suffices to let $\lambda=\mathbb{N}_{\omega}$. If $X$ is a Banach space with $dens(X)\geq\lambda$, 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 $c_0(dens (X))$.
The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1703.01891