Twisted sums of $c_0(I)$

The paper studies properties of twisted sums of a Banach space $X$ with $c_0(\kappa)$. We first prove a representation theorem for such twisted sums from which we will obtain, among others, the following: (a) twisted sums of $c_0(I)$ and $c_0(\kappa)$ are either subspaces of $\ell_\infty(\kappa)$ or trivial on a copy of $c_0(\kappa^+)$; (b) under the hypothesis $[\mathfrak p = \mathfrak c]$, when $K$ is either a suitable Corson compact, a separable Rosenthal compact or a scattered compact of finite height, there is a twisted sum of $C(K)$ with $c_0(\kappa)$ that is not isomorphic to a space of continuous functions; (c) all such twisted sums are Lindenstrauss spaces when $X$ is a Lindenstrauss space and $G$-spaces when $X=C(K)$ with $K$ convex, which shows tat a result of Benyamini is optimal; (d) they are isomorphically polyhedral when $X$ is a polyhedral space with property ($\star$), which solves a problem of Castillo and Papini.