  An equitable colouring of a graph $G$ is a colouring of the vertices of $G$
so that no two adjacent vertices are coloured the same and, additionally, the
colour class sizes differ by at most $1$. The equitable chromatic number
$\chi_=(G)$ is the minimum number of colours required for this. We study the
equitable chromatic number of the dense random graph $G(n,m)$, where $m =
\left\lfloor p {n \choose 2} \right \rfloor $ and $0<p< 0.86$ is constant. It
is a well-known question of Bollob\'as whether for $p=1/2$ there is a function
$f(n) \rightarrow \infty$ so that for any sequence of intervals of length
$f(n)$, the normal chromatic number of $G(n,m)$ lies outside the intervals with
probability at least $1/2$ if $n$ is large enough. Bollob\'as proposes that
this is likely to hold for $f(n) = \log n$. We show that for the
\emph{equitable} chromatic number, the answer to the analogous question is
negative. In fact, there is a subsequence $(n_j)_{j}$ of the integers where
$\chi_=(G(n_j,m_j)) = n/j$ with high probability, i.e., $\chi_=(G(n_j,m_j))$ is
concentrated on exactly one explicitly known value. This constitutes
surprisingly narrow concentration since in this range the equitable chromatic
number, like the normal chromatic number, is rather large in absolute value,
namely asymptotically equal to $n / (2\log_b n)$ where $b=1/(1-p)$.
