ExampleConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-26

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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Bose's construction at order $9$ gives a Steiner triple system on nine points

Example

For $m = 1$ , Bose's construction has $n = 3$ and point set $(Z / 3) \times (Z / 3)$ . Since $x \circ y = 2 (x + y)$ in $Z / 3$ , the twelve blocks are ${(0, 0), (0, 1), (0, 2)}, {(1, 0), (1, 1), (1, 2)}, {(2, 0), (2, 1), (2, 2)},$ ${(0, 0), (1, 0), (2, 1)}, {(0, 1), (1, 1), (2, 2)}, {(0, 2), (1, 2), (2, 0)},$ ${(0, 0), (2, 0), (1, 1)}, {(0, 1), (2, 1), (1, 2)}, {(0, 2), (2, 2), (1, 0)},$ ${(1, 0), (2, 0), (0, 1)}, {(1, 1), (2, 1), (0, 2)}, {(1, 2), (2, 2), (0, 0)} .$

Facts & Assumptions

Given: Bose's construction from Bose's construction gives a Steiner triple system of order $6 m + 3$ for $m \geq 1$ .

Verification

technique · direct

1.1

The first three displayed blocks are the vertical blocks $V_{x}$ , and the remaining nine are the blocks $B_{x, y, i}$ for the three unordered pairs ${0, 1}$ , ${0, 2}$ , and ${1, 2}$ and the three layers $i \in Z / 3$ .

givenalgebra

2.1

Theorem Bose's construction gives a Steiner triple system of order $6 m + 3$ for $m \geq 1$ therefore applies verbatim and shows that these twelve blocks form a Steiner triple system on nine points.

step 1.1∎

Depends on

Bose's construction gives a Steiner triple system of order $6m+3$ for $m\ge1$

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Jonathan Davidson, Steiner Triple Systems (standard reference, not scraped)

Bose's construction at order 99 gives a Steiner triple system on nine points