Alphabeta Math
TheoremStatement: AI-adaptedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-28
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.

Every perfect graph satisfies |V(G)|<=kappa(G)

Statement

If G is a perfect graph, then

∣V(G)∣≤κ(G).

Facts & Assumptions

Given: A perfect graph G.

[L1]

Perfect graphs satisfy χ(G)=ω(G) (Perfect graphs).

[L2]

Every finite graph satisfies ∣V(G)∣≤χ(G)α(G) (The bounds ω(G)≤χ(G) and ∣V(G)∣≤χ(G)α(G)).

[L3]

By definition, κ(G)=α(G)ω(G) (The parameter kappa(G)=alpha(G)omega(G), Cliques, stable sets, the clique number ω(G) and stability number α(G)).

Proof

technique · direct
1.1L1L2

Combining [L1] and [L2] gives ∣V(G)∣≤χ(G)α(G)=ω(G)α(G).

2.1step 1.1L3∎

The right-hand side of step 1.1 is κ(G) by [L3]. Therefore ∣V(G)∣≤κ(G).

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