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 independent set.

If $G$ is a graph with chromatic number $\chi(G)=m$ then must $G$ contain a subgraph $H$ with

$$\zeta(H) \gg \frac{m}{\log m}?$$