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Every point of a $2$ -design lies in the same number of blocks

Statement

Let $(P, B)$ be a $2$ - $(v, k, λ)$ design. Then there is a natural number $r$ such that every point of $P$ lies in exactly $r$ blocks.

Facts & Assumptions

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

[L1]

Every block has exactly $k$ points, with $2 \leq k < v$ and $λ \geq 1$ (A $2$ - $(v, k, λ)$ design).

[L2]

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

Proof

technique · direct

1.1

Fix a point $p \in P$ , and let $r_{p}$ be the number of blocks containing $p$ . Count the ordered pairs $(q, B)$ with $q \in P ∖ {p}$ and ${p, q} \subseteq B$ .

givenchoose

2.1

Counting by blocks through $p$ , each such block contributes $k - 1$ choices of $q$ , so the number of pairs is $r_{p} (k - 1)$ .

step 1.1L1algebra

2.2

Counting by the second point, each $q \in P ∖ {p}$ contributes exactly $λ$ blocks, so the number of pairs is $λ (v - 1)$ .

step 1.1L2algebra

3.1

Therefore $r_{p} (k - 1) = λ (v - 1)$ , so $r_{p} = λ (v - 1) / (k - 1)$ depends only on $v$ , $k$ , and $λ$ , not on $p$ . Thus every point lies in the same number $r$ of blocks.

step 2.1step 2.2algebra∎

Depends on

A $2$-$(v,k,\lambda)$ design

Used by

A 2-design satisfies bk=vr and r(k-1)=λ(v-1) Theorem

Dependency tree · one level

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Sources

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

Every point of a 22-design lies in the same number of blocks