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The standard divisibility conditions for a $2$ -design

Statement

If a $2$ - $(v, k, λ)$ design exists, then $r = \frac{λ (v - 1)}{k - 1} \in N, b = \frac{v λ (v - 1)}{k (k - 1)} \in N .$ In particular, $k - 1$ divides $λ (v - 1)$ and $k (k - 1)$ divides $v λ (v - 1)$ .

Facts & Assumptions

Given: A $2$ - $(v, k, λ)$ design.

[L1]

The counting identities are $b k = v r$ and $r (k - 1) = λ (v - 1)$ (A $2$ -design satisfies $b k = v r$ and $r (k - 1) = λ (v - 1)$ ).

Proof

technique · direct

1.1

Solving the second identity of [L1] gives $r = λ (v - 1) / (k - 1)$ , so $k - 1$ divides $λ (v - 1)$ .

L1algebra

2.1

Substituting step 1.1 into the first identity of [L1] gives $b = v λ (v - 1) / (k (k - 1))$ , so $k (k - 1)$ divides $v λ (v - 1)$ .

step 1.1L1algebra∎

Depends on

A $2$-design satisfies $bk=vr$ and $r(k-1)=\lambda(v-1)$

Used by

A Steiner triple system can exist only when v≡1 or 3 (mod 6) Corollary

Dependency tree · two levels

3 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

Noam D. Elkies, Math 155 notes: Jan. 29 (standard reference, not scraped)

The standard divisibility conditions for a 22-design