CorollaryStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)verified 2026-09-26 (gpt-6-sol)

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For every forest $H$ , graphs excluding $H$ and $\overline{H}$ have the Erdős-Hajnal property

Statement

For every forest $H$ , every finite graph with no induced $H$ and no induced $\overline{H}$ has a clique or a stable set of size at least a positive power of its order. Equivalently, the hereditary class forbidding $H$ and $\overline{H}$ has the Erdős-Hajnal property.

Facts & Assumptions

Given: A forest $H$ .

[L1]

The class of graphs with no induced $H$ and no induced $\overline{H}$ has the strong Erdős-Hajnal property (For every forest $H$ , graphs excluding $H$ and $\overline{H}$ have a linear pure pair).

[L2]

Every class defined by forbidden induced subgraphs is hereditary (Every class defined by forbidden induced subgraphs is hereditary).

[L3]

Every hereditary class with the strong Erdős-Hajnal property has the Erdős-Hajnal property (The strong Erdős–Hajnal property implies the Erdős–Hajnal property).

Proof

technique · direct

1.1

Let $C_{H}$ be the class of finite graphs with no induced $H$ and no induced $\overline{H}$ . By [L2], this is a hereditary class, and by [L1] it has the strong Erdős-Hajnal property.

L1L2

2.1

Applying [L3] to $C_{H}$ gives the Erdős-Hajnal property for that class. This is the claimed graph-level statement.

step 1.1L3∎

Depends on

Used by

The leaf/co-leaf corollary recovers the P₅ case from the P₄ case Example

Dependency tree · two levels

15 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, Alex Scott, Paul Seymour, and Sophie Spirkl, Pure pairs. I. Trees and linear anticomplete pairs, statement 1.3 (standard reference, not scraped)

For every forest HH, graphs excluding HH and H‾H have the Erdős-Hajnal property