DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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.

Homogeneous vertex sets and the homogeneous number $hom (G) = \max {ω (G), α (G)}$

Definition

Let $G$ be a finite graph. A vertex set $S \subseteq V (G)$ is homogeneous if it is a clique or a stable set in $G$ (Cliques, stable sets, the clique number $ω (G)$ and stability number $α (G)$ ). The homogeneous number of $G$ is $hom (G) : = \max {ω (G), α (G)} .$

For the null graph, the published conventions give $ω (G) = α (G) = 0$ , and hence $hom (G) = 0$ .

Depends on

Cliques, stable sets, the clique number $\omega(G)$ and stability number $\alpha(G)$

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 30 results over 13 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

M. Chudnovsky, The Erdos-Hajnal Conjecture: A Survey, sec. 1 (standard reference, not scraped)
A. Chernikov, MATH 223M notes, sec. 3.1 (standard reference, not scraped)

Homogeneous vertex sets and the homogeneous number hom⁡(G)=max⁡{ω(G),α(G)}hom(G)=max{ω(G),α(G)}