CorollaryStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-08-31

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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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The five-vertex path and its complement have the Erdős-Hajnal property

Statement

Both $P_{5}$ and $\overline{P_{5}}$ have the Erdős-Hajnal property.

Facts & Assumptions

Given: The graph $P_{5}$ .

[L1]

The graph $P_{5}$ has the polynomial Rödl property (The five-vertex path has the polynomial Rödl property).

[L2]

Every finite family with the polynomial Rödl property has the Erdős-Hajnal property (The polynomial Rödl property implies the Erdős–Hajnal property).

[F1]

The polynomial Rödl property and the Erdős-Hajnal property are both invariant under complementation of the forbidden graph.

Proof

technique · direct

1.1

Applying [L2] to the singleton family ${P_{5}}$ and using [L1], we conclude that $P_{5}$ has the Erdős-Hajnal property.

L1L2

2.1

By [F1], the same holds for $\overline{P_{5}}$ .

step 1.1

3.1

Therefore both $P_{5}$ and $\overline{P_{5}}$ have the Erdős-Hajnal property.

step 1.1step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

20 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

Tung Nguyen, Alex Scott, and Paul Seymour, Induced subgraph density. VII. The five-vertex path, Theorem 1.2 (standard reference, not scraped)
Shenwei Huang, Yiao Ju, and Yidong Zhou, Erdős-Hajnal beyond the five-vertex path, Theorem 1.6 (standard reference, not scraped)