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A $2$ -design satisfies $b k = v r$ and $r (k - 1) = λ (v - 1)$

Statement

Let $(P, B)$ be a $2$ - $(v, k, λ)$ design. Let $b : = ∣ B ∣$ , and let $r$ be the common number of blocks through a point. Then $b k = v r, r (k - 1) = λ (v - 1) .$

Facts & Assumptions

Given: A $2$ - $(v, k, λ)$ design $(P, B)$ .

[L1]

Every point of the design lies in the same number $r$ of blocks (Every point of a $2$ -design lies in the same number of blocks).

[L2]

Every block has exactly $k$ points and every two-element subset of $P$ lies in exactly $λ$ blocks (A $2$ - $(v, k, λ)$ design).

Proof

technique · direct

1.1

Let $b : = ∣ B ∣$ , and let $r$ be the common number of blocks through a point from [L1].

L1choose

2.1

Count the incident pairs $(p, B)$ with $p \in B$ . Each block contributes $k$ such pairs, so the total is $b k$ ; each of the $v$ points contributes $r$ such pairs, so the total is also $v r$ . Hence $b k = v r$ .

step 1.1L2algebra

3.1

Fix a point $p \in P$ and count the ordered pairs $(q, B)$ with $q \in P ∖ {p}$ and ${p, q} \subseteq B$ . By [L2], each of the $r$ blocks through $p$ contributes $k - 1$ choices of $q$ , while each of the $v - 1$ other points contributes exactly $λ$ blocks. Therefore $r (k - 1) = λ (v - 1)$ .

step 1.1L2algebra∎

Depends on

Used by

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Sources

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

A 22-design satisfies bk=vrbk=vr and r(k−1)=λ(v−1)r(k−1)=λ(v−1)