DefinitionDefinition: AI-adaptedProof: Not applicableaudited 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 $Ω$ . The Petersen graph has vertex set $[Ω]^{2}$ , the set of two-element subsets of $Ω$ , and edge set

${{A, B} \subseteq [Ω]^{2} : ∣ {A, B} ∣ = 2 and A \cap B = \emptyset} .$

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 $(\binom{5}{2}) = 10$ elements by A finite set with $n$ elements has exactly $(\binom{n}{2})$ two-element subsets, and $2 (\binom{n}{2}) = n (n - 1)$ . The definition is independent of the names assigned to the five elements, because a relabelling preserves disjointness.

Depends on

Used by

Dependency tree · two levels

15 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

R. Diestel, Graph Theory, Chapter 1 preview (standard reference, not scraped)