PropositionStatement: AI-adaptedProof: AI-generatedprecheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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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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If $H$ is an induced subgraph of $H^{'}$ and $H^{'}$ has the Erdős–Hajnal property, then $H$ has it with every constant of $H^{'}$

Statement

Suppose $H$ has an induced embedding into $H^{'}$ . Then every Erdős–Hajnal constant for $H^{'}$ is one for $H$ . Consequently, if $H^{'}$ has the Erdős–Hajnal property, then so does $H$ .

Facts & Assumptions

Given: Finite graphs $H, H^{'}$ and an induced embedding $H ↪_{i n d} H^{'}$ .

[L1]

Every Erdős–Hajnal constant passes from a hereditary class to any hereditary subclass (The Erdős–Hajnal property and each of its constants pass to hereditary subclasses).

[L2]

A graph is $F$ -free when it has no induced embedding of $F$ ( $H$ -free and $F$ -free graphs under the induced-subgraph convention).

[L3]

Induced embeddings compose, so induced-subgraph containment is transitive (Induced embeddings compose, and the induced-subgraph relation is transitive up to isomorphism).

[L4]

Every fixed-pattern-free graph class is hereditary (Every class defined by forbidden induced subgraphs is hereditary).

Proof

technique · direct

1.1givenL2L3

If $G$ is $H$ -free, then it is $H^{'}$ -free: an induced embedding $H^{'} ↪_{i n d} G$ would compose with the Given embedding to put $H$ inducedly in $G$ .

2.1step 1.1L4

Hence the $H$ -free class is a subclass of the $H^{'}$ -free class, and both are hereditary by [L4].

3.1step 2.1L1∎

Applying [L1] proves that every constant of $H^{'}$ is a constant of $H$ , and therefore proves the property implication.

Depends on

Used by

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Dependency tree · two levels

11 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

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

If HH is an induced subgraph of H′H′ and H′H′ has the Erdős–Hajnal property, then HH has it with every constant of H′H′