PropositionStatement: AI-adaptedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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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.
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The Erdős–Hajnal property and each of its constants pass to hereditary subclasses

Statement

If $D \subseteq C$ are hereditary graph classes, then every Erdős–Hajnal constant for $C$ is one for $D$ . In particular, the Erdős–Hajnal property passes from $C$ to $D$ .

Facts & Assumptions

Given: Hereditary graph classes $D \subseteq C$ and an Erdős–Hajnal constant $ϵ$ for $C$ .

[L1]

The constant condition says that every nonempty $G$ in the class satisfies $hom (G) \geq ∣ V (G) ∣^{ϵ}$ (The Erdős–Hajnal property and an Erdős–Hajnal constant for a hereditary graph class).

Proof

technique · direct

1.1

Every nonempty $G \in D$ also lies in $C$ , so [L1] gives $hom (G) \geq ∣ V (G) ∣^{ϵ}$ .

givenL1

2.1

Thus $ϵ$ is a constant for $D$ ; the existence assertion follows by retaining any constant of $C$ .

step 1.1L1∎

Depends on

The Erdős–Hajnal property and an Erdős–Hajnal constant for a hereditary graph class

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 9 results over 4 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

Erdos-Hajnal properties in graphs and hypergraphs, introduction (standard reference, not scraped)