  Let $Q(n,c)$ denote the minimum clique number over graphs with $n$ vertices
and chromatic number $c$. We determine the rate of growth of of the sequence
${Q(n,\lceil rn \rceil)}_{n=1}^\infty$ for any fixed $0<r\leq 1$. We also give
a better upper bound for $Q(n,\lceil rn \rceil)$.
