DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (gpt-5.6-terra)audited 2026-08-26

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

A $2$ - $(v, k, λ)$ design

Definition

Let $v, k, λ$ be natural numbers with $2 \leq k < v$ and $λ \geq 1$ . A $2$ - $(v, k, λ)$ design is a pair $(P, B)$ such that:

$P$ is a finite set with $∣ P ∣ = v$ ;
$B$ is a finite collection of distinct $k$ -element subsets of $P$ , called the blocks;
every subset of $P$ with cardinality $2$ lies in exactly $λ$ blocks.

Remarks

The number of blocks and the number of blocks through a point are not part of the definition. They are derived immediately below.

Used by

A projective plane of order n is a symmetric 2-(n²+n+1,n+1,1) design Corollary
A symmetric design Definition
Steiner triple systems Definition
The point-block incidence matrix of a 2-design Definition
Every point of a 2-design lies in the same number of blocks Lemma
A 2-design satisfies bk=vr and r(k-1)=λ(v-1) Theorem
A Steiner triple system exists exactly for orders v>3 with v≡1 or 3 (mod 6) Theorem
For a 2-design, NN^T=(r-λ)I+λ J Theorem

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

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