A \(\delta \) -colouring of the point set of a block design is said to be weak if no block is monochromatic. The 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 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 chromatic number 3. Moreover, for each integer \(\delta \geqslant 3\) there exist infinitely many Steiner triple systems with chromatic number \(\delta \) . We consider colourings of the subclass of Steiner triple systems which are resolvable. A Kirkman triple system consists of a resolvable Steiner triple system together with a partition of its blocks into parallel classes. We show that for each \(v\equiv 3\pmod {6}\) there exists a Kirkman triple system on v points with chromatic number 3. We also show that for each integer \(\delta \geqslant 3\) , there exist infinitely many Kirkman triple systems with chromatic number \(\delta \) . We close with several open problems.
