TheoremStatement: Literature-sourcedProof: AI-adapted

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A Steiner triple system exists exactly for orders $v > 3$ with $v \equiv 1$ or $3 (m o d 6)$

Statement

A Steiner triple system of order $v$ exists if and only if $v > 3$ and $v \equiv 1$ or $3 (m o d 6)$ .

Facts & Assumptions

Given: A natural number $v$ .

[L1]

If a Steiner triple system of order $v$ exists, then $v \equiv 1$ or $3 (m o d 6)$ (A Steiner triple system can exist only when $v \equiv 1$ or $3 (m o d 6)$ ).

[L2]

For every natural number $m \geq 1$ , Bose's construction yields a Steiner triple system of order $6 m + 3$ (Bose's construction gives a Steiner triple system of order $6 m + 3$ for $m \geq 1$ ).

[L3]

For every integer $m \geq 1$ , Skolem's construction yields a Steiner triple system of order $6 m + 1$ (Skolem's construction gives a Steiner triple system of order $6 m + 1$ ).

[L4]

A Steiner triple system of order $v$ is a $2$ - $(v, 3, 1)$ design, and a $2$ -design requires $2 \leq 3 < v$ (Steiner triple systems, A $2$ - $(v, k, λ)$ design).

Proof

technique · direct

1.1L1L4

If a Steiner triple system of order $v$ exists, then [L4] gives $v > 3$ , and [L1] gives $v \equiv 1$ or $3 (m o d 6)$ .

1.2L2algebra

Conversely, suppose $v > 3$ and $v \equiv 3 (m o d 6)$ . Then $v = 6 m + 3$ for some natural number $m$ , and $v > 3$ forces $m \geq 1$ . So [L2] gives a Steiner triple system of order $v$ .

1.3L3algebra

Suppose instead that $v > 3$ and $v \equiv 1 (m o d 6)$ . Then $v = 6 m + 1$ for some integer $m \geq 1$ , because $v \equiv 1 (m o d 6)$ and $v > 3$ force $v \geq 7$ . Now [L3] gives a Steiner triple system of order $v$ .

2.1step 1.1step 1.2step 1.3∎

The two congruence classes $1$ and $3$ modulo $6$ exhaust the converse assumption, so steps 1.2 and 1.3 prove that direction. Together with step 1.1, this proves the theorem.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

12 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)

A Steiner triple system exists exactly for orders v>3v>3 with v≡1v≡1 or 3(mod6)3(mod6)