  This paper is concerned with the isomorphic structure of the Banach space
$\ell_\infty/c_0$ and how it depends on combinatorial tools whose existence is
consistent but not provable from the usual axioms of ZFC. Our main global
result is that it is consistent that $\ell_\infty/c_0$ does not have an
orthogonal \break $\ell_\infty$-decomposition that is, it is not of the form
$\ell_\infty(X)$ for any Banach space $X$. The main local result is that it is
consistent that $\ell_\infty(c_0(\mathfrak{c}))$ does not embed isomorphically
into $\ell_\infty/c_0$, where $\mathfrak{c}$ is the cardinality of the
continuum, while $\ell_\infty$ and $c_0(\mathfrak{c})$ always do embed quite
canonically. This should be compared with the results of Drewnowski and Roberts
that under the assumption of the continuum hypothesis $\ell_\infty/c_0$ is
isomorphic to its $\ell_\infty$-sum and in particular it contains an isomorphic
copy of all Banach spaces of the form $\ell_\infty(X)$ for any subspace $X$ of
$\ell_\infty/c_0$.
