Corollary B: no Steiner triple system on $9$ points has a role colouring

Theorem no_role_colouring_nine. The theorem says that no Steiner triple system on 9 points has a "role colouring", whatever role assignment is chosen.

Objects. Let $V = \{0,1,\dots,8\}$ be the 9-element point set.

• The system $S$. $S$ is a Steiner triple system on $V$, given by a finite set $\mathcal{L}$ of subsets of $V$ called lines. Because $\mathcal{L}$ is a set, the same line cannot occur twice. The system has two properties:

  • every line $\ell \in \mathcal{L}$ has exactly 3 elements;
  • for any two distinct points $x \neq y$ in $V$, exactly one line $\ell \in \mathcal{L}$ contains both $x$ and $y$.

  There are no other conditions.

• The role function. $\mathrm{role} : V \times \mathcal{P}(V) \to \{0,1,2\}$ is an arbitrary function. It gives a value $\mathrm{role}(x,\ell)$ for every point $x$ and every subset $\ell \subseteq V$, including subsets that are not lines and pairs where $x \notin \ell$. The definition below only looks at the values $\mathrm{role}(x,\ell)$ where $\ell \in \mathcal{L}$ and $x \in \ell$.

The expanded definition. A pair $(S, \mathrm{role})$ is a role colouring when all three of these conditions hold:

1. Injective on each line. For every line $\ell \in \mathcal{L}$ and all points $x, y \in \ell$:

Each line has 3 points and there are 3 values, so the 3 points of every line get the 3 values $0,1,2$, each exactly once. 2. Every value appears at every point. For every point $x \in V$ and every value $\rho \in \{0,1,2\}$, some line $\ell \in \mathcal{L}$ satisfies $x \in \ell$ and $\mathrm{role}(x,\ell) = \rho$. 3. Injective at each point. For every point $x \in V$ and all lines $\ell, \ell' \in \mathcal{L}$ with $x \in \ell$ and $x \in \ell'$:

Conditions 2 and 3 together say that, for each point $x$, the map $\ell \mapsto \mathrm{role}(x,\ell)$ is a bijection from the lines through $x$ onto $\{0,1,2\}$. In particular, each point lies on exactly 3 lines.

Statement. For every Steiner triple system $S$ on the 9 points $V$ (as described above) and every function $\mathrm{role} : V \times \mathcal{P}(V) \to \{0,1,2\}$:

In other words, for every such $S$ and $\mathrm{role}$, at least one of conditions 1–3 fails.

Quantifiers and edge cases.

• The statement is universally quantified over all systems $S$ and all functions $\mathrm{role}$, with no further hypotheses.
• Such systems do exist on 9 points; the affine plane of order 3 is one, with 12 lines. So the quantifier over $S$ is not empty.
• In any such system, each point lies on exactly 4 lines. The lines through a point $x$ split the other 8 points into pairs, so there are $8/2 = 4$ of them.
• The statement is fixed to $n = 9$ points. It says nothing about other values of $n$.