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

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In a symmetric $2$ -design, distinct blocks meet in exactly $λ$ points

Statement

Let $(P, B)$ be a symmetric $2$ - $(v, k, λ)$ design. Then every two distinct blocks of $B$ meet in exactly $λ$ points.

Facts & Assumptions

Given: A symmetric $2$ - $(v, k, λ)$ design with incidence matrix $N$ .

[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 $r (k - 1) = λ (v - 1)$ ).

[L2]

The incidence identity is $N N^{T} = (r - λ) I_{v} + λ J_{v}$ (For a $2$ -design, $N N^{T} = (r - λ) I + λ J$ ).

Proof

technique · direct

1.1

Symmetry gives $b = v$ , so the first identity in [L1] yields $r = k$ .

L1algebra

2.1

Using step 1.1 in the second identity of [L1], one gets $k (k - 1) = λ (v - 1)$ , so $k - λ = k (v - k) / (v - 1) > 0$ because $k < v$ .

step 1.1L1algebra

3.1

If $x^{T} N = 0$ , then $x^{T} N N^{T} x = 0$ , but [L2] and step 2.1 give $x^{T} N N^{T} x = (k - λ) \sum_{i} x_{i}^{2} + λ (\sum_{i} x_{i})^{2}$ , which is positive for every nonzero $x$ . Hence $N$ is invertible.

step 2.1L2algebra

4.1

Every row and every column of $N$ has sum $k$ : rows because each point lies in $r = k$ blocks by step 1.1, and columns because every block has size $k$ . Therefore $N J_{v} = J_{v} N = k J_{v}$ , so step 3.1 gives $N^{- 1} J_{v} N = J_{v}$ .

step 1.1step 3.1L1algebra

5.1

Multiplying the identity of [L2] on the left by $N^{- 1}$ and on the right by $N$ gives $N^{T} N = (k - λ) I_{v} + λ J_{v}$ .

step 1.1step 4.1L2algebra

6.1

The $(B, C)$ entry of $N^{T} N$ counts the points in $B \cap C$ , so step 5.1 shows that every off-diagonal entry is $λ$ . Thus distinct blocks meet in exactly $λ$ points.

step 5.1algebra∎

Depends on

Used by

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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

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

In a symmetric 22-design, distinct blocks meet in exactly λλ points