TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-26

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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Fisher's inequality: every $2$ - $(v, k, λ)$ design has at least $v$ blocks

Statement

Every $2$ - $(v, k, λ)$ design has at least $v$ blocks.

Facts & Assumptions

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

[L1]

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

[L2]

The counting identities give $r (k - 1) = λ (v - 1)$ , with $2 \leq k < v$ and $λ \geq 1$ (A $2$ -design satisfies $b k = v r$ and $r (k - 1) = λ (v - 1)$ ).

Proof

technique · direct

1.1L2algebra

From [L2] one gets $r - λ = λ (v - k) / (k - 1) > 0$ , because $λ \geq 1$ and $v > k$ .

2.1step 1.1L1algebra

If $x \in R^{v}$ is nonzero, then [L1] gives $x^{T} N N^{T} x = (r - λ) \sum_{i} x_{i}^{2} + λ (\sum_{i} x_{i})^{2} > 0$ by step 1.1. Therefore no nonzero vector satisfies $x^{T} N = 0$ .

3.1step 2.1algebra∎

So the $v$ rows of $N$ are linearly independent in $R^{b}$ . A family of $v$ linearly independent vectors in $R^{b}$ requires $v \leq b$ .

Remarks

The positivity argument is over $R$ . The published false statement FALSE: distinct nonempty $A_{1}, \dots, A_{m} \subseteq [n]$ whose pairwise intersections all have the same parity satisfy $m \leq n$ records why the same proof does not survive over $F_{2}$ .

Depends on

Used by

The Fano plane has incidence matrix N with NN^T=2I+J Example

Dependency tree · two levels

5 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)

Fisher's inequality: every 22-(v,k,λ)(v,k,λ) design has at least vv blocks