The cochromatic number of $G$, denoted by $\zeta(G)$, is the minimum number of colours needed to colour the vertices of $G$ such that each colour class induces either a complete graph or empty graph. Let $z(n)$ be the maximum value of $\zeta(G)$ over all graphs $G$ with $n$ vertices.

Determine $z(n)$ for small values of $z(n)$. In particular is it true that $z(12)=4$?