Alphabeta Math
DefinitionDefinition: AI-adaptedProof: Not applicableSession-authored (Fable 5 assisted)audited 2026-07-31
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.

The Petersen graph on the two-element subsets of a five-element set, adjacent when disjoint

Definition

Fix a five-element set Ω\Omega. The Petersen graph has vertex set [Ω]2[\Omega]^2, the set of two-element subsets of Ω\Omega, and edge set

{ {A,B}⊆[Ω]2:∣{A,B}∣=2 and A∩B=∅ }.\{\,\{A,B\}\subseteq [\Omega]^2:|\{A,B\}|=2\text{ and }A\cap B=\varnothing\,\}.

In words, two vertices are adjacent exactly when the corresponding two-element subsets are disjoint. This is a finite simple graph, and its vertex set has (52)=10\binom52=10 elements by A finite set with nn elements has exactly (n2)\binom{n}{2} two-element subsets, and 2(n2)=n(n−1)2\binom{n}{2} = n(n-1). The definition is independent of the names assigned to the five elements, because a relabelling preserves disjointness.

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 48 results over 16 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources