The role postulates force the Fano plane (C1 Theorem 3.6)

roles_force_fano. Let $n$ be a natural number with $n > 0$, and write $[n] = \{0, 1, \dots, n-1\}$ for the point set.

The system $S$. Let $S$ be a Steiner triple system on $[n]$ in the following sense: $S$ is a finite family $\mathcal{L}$ of subsets of $[n]$ ("lines") such that

• every line $l \in \mathcal{L}$ has exactly $3$ elements, and
• for every two distinct points $x \neq y$ in $[n]$ there is exactly one line $l \in \mathcal{L}$ with $x \in l$ and $y \in l$.

No other conditions are imposed. There is no requirement that $\mathcal{L}$ be nonempty or that every point lie on a line; for instance, for $n = 1$ the empty family is such a system.

The role function. Let $\mathrm{role}$ be an arbitrary function that assigns to each point $x \in [n]$ and each subset $A \subseteq [n]$ a value

$$\mathrm{role}(x, A) \in \{0, 1, 2\}.$$

It is defined on all pairs (point, subset), but only its values at pairs $(x, l)$ with $l$ a line and $x \in l$ appear in the hypothesis below.

Hypothesis ("role colouring"). Assume all three of the following:

1. Distinct roles within a line: for every line $l \in \mathcal{L}$ and all $x, y \in l$,

$$\mathrm{role}(x, l) = \mathrm{role}(y, l) \implies x = y.$$

2. Every role occurs at every point: for every point $x \in [n]$ and every $\rho \in \{0,1,2\}$ there is a line $l \in \mathcal{L}$ with $x \in l$ and $\mathrm{role}(x, l) = \rho$.
3. Distinct roles at a point: for every point $x \in [n]$ and all lines $l, l' \in \mathcal{L}$ with $x \in l$ and $x \in l'$,

$$\mathrm{role}(x, l) = \mathrm{role}(x, l') \implies l = l'.$$

Taken together, (2) and (3) say that for each point $x$, the map $l \mapsto \mathrm{role}(x, l)$ is a bijection from the lines through $x$ onto $\{0,1,2\}$. Condition (1) says that on each line, $x \mapsto \mathrm{role}(x, l)$ is injective. In the degenerate case $n = 1$ the only admissible $S$ has no lines, and then (2) cannot hold, so the hypothesis is unsatisfiable for $n = 1$. The case $n = 0$ is excluded by $n > 0$.

Conclusion ("$S$ is the Fano plane"). There exists a bijection $e : [n] \to \{0, 1, \dots, 6\}$ such that the family of images of the lines under $e$ is exactly the Fano line set:

$$\{\, e(l) : l \in \mathcal{L} \,\} = \{\, F_i : i \in \mathbb{Z}/7 \,\}, \qquad F_i = \{\, i,\ i+1,\ i+3 \,\} \ (\text{arithmetic mod } 7).$$

Explicitly, the seven Fano lines are

$$\{0,1,3\},\ \{1,2,4\},\ \{2,3,5\},\ \{3,4,6\},\ \{4,5,0\},\ \{5,6,1\},\ \{6,0,2\}.$$

This is an equality of families of sets. Because a bijection $[n] \to \{0,\dots,6\}$ exists, the conclusion in particular forces $n = 7$.

In summary, the statement asserts: for every $n > 0$, every Steiner triple system $S$ on $[n]$ as above, and every role function satisfying (1)–(3), the number of points is $n = 7$ and $S$ is isomorphic, via a relabelling of the points, to the Fano plane with lines $\{i, i+1, i+3\} \bmod 7$.