TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-09-26

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Every $H$ -free graph has a polynomial-size pure pair

Statement

For every finite graph $H$ , there exists a real constant $c_{H} > 0$ such that every finite $H$ -free graph $G$ with $∣ V (G) ∣ \geq 2$ contains disjoint sets $A, B \subseteq V (G)$ satisfying

$∣ A ∣ \geq ⌊ ∣ V (G) ∣^{c_{H}} ⌋, ∣ B ∣ \geq ⌊ ∣ V (G) ∣^{c_{H}} ⌋,$

and such that $(A, B)$ is a pure pair.

Facts & Assumptions

Given: A finite graph $H$ and a finite $H$ -free graph $G$ with $∣ V (G) ∣ \geq 2$ .

[F1]

The Erdős-Hajnal-Pach theorem gives a real constant $δ_{H} > 0$ such that every finite $H$ -free graph on at least two vertices contains disjoint sets $A, B \subseteq V (G)$ with $∣ A ∣ \geq ∣ V (G) ∣^{δ_{H}}, ∣ B ∣ \geq ∣ V (G) ∣^{δ_{H}},$ and with $A$ complete or anticomplete to $B$ .

[L1]

A pair of disjoint vertex sets is pure exactly when it is complete or anticomplete (Edges between disjoint vertex sets; complete, anticomplete, pure and mixed pairs).

Proof

technique · direct

1.1

By [F1], choose $δ_{H} > 0$ and disjoint sets $A, B \subseteq V (G)$ with $∣ A ∣ \geq ∣ V (G) ∣^{δ_{H}}, ∣ B ∣ \geq ∣ V (G) ∣^{δ_{H}},$ such that $A$ is complete or anticomplete to $B$ . Set $c_{H} : = δ_{H}$ . Then $∣ A ∣$ and $∣ B ∣$ are certainly at least $⌊ ∣ V (G) ∣^{c_{H}} ⌋$ .

F1choosealgebra

2.1

By [L1], the pair $(A, B)$ is pure. Therefore the constant $c_{H}$ has the required property.

step 1.1L1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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, Theorem 3.1 (standard reference, not scraped)

Every HH-free graph has a polynomial-size pure pair