DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge 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.

A good function on a graph

Definition

Let $G$ be a finite graph. A function $g : V (G) \to R_{\geq 0}$ is good for $G$ when

$\sum_{v \in V (P)} g (v) \leq 1$

for every perfect induced subgraph $P$ of $G$ (A perfect graph).

Thus a good function is a nonnegative vertex-weighting whose total weight on each perfect induced subgraph is at most $1$ .

Depends on

A perfect graph

Used by

An α-narrow graph Definition
The five-cycle is 2-narrow but not 1-narrow Example
Every perfect graph is 1-narrow Proposition
An α-narrow graph contains a perfect induced subgraph of order at least |V(G)|^1/α Theorem

Dependency tree · two levels

4 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

Maria Chudnovsky, The Erdős-Hajnal Conjecture: A Survey, Section 2 (standard reference, not scraped)
Maria Chudnovsky and Shmuel Safra, The Erdős-Hajnal conjecture for bull-free graphs, Section 2 (standard reference, not scraped)