  A $\delta$-colouring of the point set of a block design is said to be {\em
weak} if no block is monochromatic. The {\em weak chromatic number} $\chi(S)$
of a block design $S$ is the smallest integer $\delta$ such that $S$ has a weak
$\delta$-colouring. It has previously been shown that any Steiner triple system
has weak chromatic number at least $3$ and that for each $v\equiv 1$ or
$3\pmod{6}$ there exists a Steiner triple system on $v$ points that has weak
chromatic number $3$. Moreover, for each integer $\delta \geq 3$ there exist
infinitely many Steiner triple systems with weak chromatic number $\delta$.
  We consider colourings of the subclass of Steiner triple systems which are
resolvable. We show that for each $v\equiv 3\pmod{6}$ there exists a Kirkman
triple system on $v$ points with weak chromatic number $3$. We also show that
for each integer $\delta \geq 3$, there exist infinitely many Kirkman triple
systems with weak chromatic number $\delta$. We close with several open
problems.
