Weak colourings of Kirkman triple systems

A δ-colouring of the point set of a block design is said to be { weak} if no block is monochromatic. The { weak chromatic number} χ(S) of a block design S is the smallest integer δ such that S has a weak δ-colouring. It has previously been shown that any Steiner triple system has weak chromatic number at least $3$ and that for each v≡ 1 or 36 there exists a Steiner triple system on v points that has weak chromatic number $3$. Moreover, for each integer δ ≥ 3 there exist infinitely many Steiner triple systems with weak chromatic number δ. We consider colourings of the subclass of Steiner triple systems which are resolvable, namely Kirkman triple systems. We show that for each v≡ 36 there exists a Kirkman triple system on v points with weak chromatic number $3$. We also show that for each integer δ ≥ 3, there exist infinitely many Kirkman triple systems with weak chromatic number δ. We close with several open problems.