Weak colourings of Kirkman triple systems

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Problem: For each integer $\delta\ge 3$, does there exist a $\delta$-chromatic quadruple system $Q$, i.e. a $2\!-\!(v,4,1)$ design with weak chromatic number $\chi(Q)=\delta$, and some Kirkman triple system $K(Q)$ obtained from $Q$ by the standard doubling construction, such that $\chi(K(Q))=\delta$?

Result: No. The statement is already false for $\delta=3$.

Let $Q$ be any quadruple system and let $K(Q)$ be any Kirkman triple system obtained from the construction in the paper. I claim:

Hence if $Q$ is $3$-chromatic, then every $K(Q)$ satisfies $\chi(K(Q))\ge 4$, so $\chi(K(Q))\ne 3$.

Proof. Suppose $K(Q)$ has a weak 3-colouring. Rename colours so that $\infty$ has colour $0$. For each point $x\in Q$, let $x'$ be its primed copy in $K(Q)$. Since $\{\infty,x,x'\}$ is a triple, $x$ and $x'$ are not both colour $0$.

Define a 2-colouring $\theta$ of the points of $Q$ by

We show $\theta$ weakly 2-colours $Q$. Let $B=\{w,x,y,z\}$ be a block of $Q$. In $K(Q)$, the nine points

carry a copy of the unique $KTS(9)$, with the four triples through $\infty$ equal to $\{\infty,u,u'\}$, $u\in B$.

If all four points of $B$ had $\theta=1$, then this local $KTS(9)$ would use only colours $0,1$, giving a weak 2-colouring of an $STS(9)$, impossible. Indeed, in a weak 2-colouring of an $STS(9)$, if the colour classes have sizes $s,9-s$, then each of the 12 triples contributes exactly one same-colour pair, so

which has no integer solution.

If all four points of $B$ had $\theta=0$, then from each pair $\{u,u'\}$ choose a point of colour $2$. In the affine-plane model of the unique $STS(9)$, with $\infty$ as the origin, the four pairs through $\infty$ are $\{\pm v\}$ over the four one-dimensional subspaces of $\mathbb F_3^2$. Any transversal choosing one point from each pair contains an affine line, hence a triple of the local $KTS(9)$. That triple would be monochromatic of colour $2$, contradiction.

Thus no block $B$ of $Q$ is monochromatic under $\theta$, so $\chi(Q)\le 2$. Therefore no 3-chromatic quadruple system $Q$ can satisfy $\chi(K(Q))=3$.

So Question 7.5, interpreted as asking this for every $\delta\ge3$, has a negative answer.

Citation: Definitions and the construction $K(Q)$ are from Burgess, Cavenagh, Danziger, and Pike, “Weak colourings of Kirkman triple systems,” arXiv:2404.06066, Section 6. The disproof above is not taken from a cited source.