LemmaStatement: AI-adaptedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-26

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If $ϵ$ is an Erdős–Hajnal constant for $H$ and $W$ is a nonempty vertex set with $∣ W ∣^{ϵ} > hom (G)$ , then $G [W]$ has an induced copy of $H$

Statement

Let $H$ be a finite simple graph and let $ϵ > 0$ be an Erdős–Hajnal constant for the hereditary class of $H$ -free graphs. Let $G$ be a finite simple graph and let $W \subseteq V (G)$ be nonempty with $∣ W ∣^{ϵ} > hom (G)$ . Then $G [W]$ has an induced copy of $H$ .

Facts & Assumptions

Given: A finite simple graph $H$ , an Erdős–Hajnal constant $ϵ > 0$ for the class of $H$ -free graphs, a finite simple graph $G$ , and a nonempty $W \subseteq V (G)$ with $∣ W ∣^{ϵ} > hom (G)$ .

[F1]

A real $ϵ > 0$ is an Erdős–Hajnal constant for a hereditary class $C$ when every nonempty $J \in C$ satisfies $hom (J) \geq ∣ V (J) ∣^{ϵ}$ (The Erdős–Hajnal property and an Erdős–Hajnal constant for a hereditary graph class, Real powers for positive bases, with the zero-base positive-exponent convention).

[F2]

$G$ is $H$ -free when $G$ has no induced copy of $H$ ( $H$ -free and $F$ -free graphs under the induced-subgraph convention, Induced embeddings and induced copies of a graph).

[L1]

For every family $F$ of finite graphs, the class of $F$ -free finite graphs is hereditary (Every class defined by forbidden induced subgraphs is hereditary).

[L2]

$hom (G [W]) \leq hom (G)$ for every $W \subseteq V (G)$ ( $hom (G [W]) \leq hom (G)$ for every vertex subset $W$ ).

[F3]

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

[F4]

$G [W]$ has vertex set $W$ (Subgraphs, induced subgraphs and spanning subgraphs).

Proof

technique · contrapositive

1.1

It suffices to prove the contrapositive: if $G [W]$ has no induced copy of $H$ , then $∣ W ∣^{ϵ} \leq hom (G)$ .

contrapositive-reduce

1.2

Assume $G [W]$ has no induced copy of $H$ . Then $G [W]$ is $H$ -free, and the class of $H$ -free graphs is hereditary, so $G [W]$ is a member of the class for which $ϵ$ is an Erdős–Hajnal constant.

assume-hypF2L1

2.1

The graph $G [W]$ is nonempty, since $W \neq \emptyset$ and $V (G [W]) = W$ , so [F1] applies to it and gives $hom (G [W]) \geq ∣ W ∣^{ϵ}$ .

step 1.2F1F3F4given

3.1

By [L2] we have $hom (G) \geq hom (G [W])$ , so $hom (G) \geq ∣ W ∣^{ϵ}$ , which is the conclusion of the contrapositive; the Statement follows.

step 2.1L2discharge-contrapositive∎

Depends on

Used by

Alon–Pach–Solymosi: if H₁ and H₂ have the Erdős–Hajnal property, so does the graph obtained from H₁ by substituting H₂ for a vertex Theorem

Dependency tree · two levels

17 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 Erdős–Hajnal Conjecture: A Survey, sec. 2 (standard reference, not scraped)

If ϵϵ is an Erdős–Hajnal constant for HH and WW is a nonempty vertex set with ∣W∣ϵ>hom⁡(G)∣W∣ϵ>hom(G), then G[W]G[W] has an induced copy of HH