Problem 1.Characterize compact Hausdorff spaces $K$ for which the Banach space $C(K)$ of continuous real-valued functions embeds into the Banach space $c_0$.

Since $c_0$ has separable dual, such $K$ must me countable. So, we can make Problem 1 more precise:

Problem 2.Is it true that for every compact countable space $K$ the Banach space $C(K)$ is isomorphic to a subspace of $c_0$?

Another possible option:

Problem 3.Let $K$ be a compact Hausdorff space. Is it true that the Banach space $C(K)$ is isomorphic to $c_0$ if $C(K)$ is isomorphic to a subspace of $c_0$?