The difference between the chromatic and the cochromatic number of a random graph

The cochromatic number $\zeta (G)$ of a graph $G$ is the minimum number of colours needed for a vertex colouring where every colour class is either an independent set or a clique. Let $\chi (G)$ denote the usual chromatic number. Around 1991 Erdős and Gimbel asked: For the random graph $G\sim G_{n,1\mathrm{/}2}$, does $\chi (G)-\zeta (G)\to \mathrm{\infty }$ whp? Erdős offered $100 for a positive and $1,000 for a negative answer. We give a positive answer to this question for roughly 95% of all values $n$.