CorollaryStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-31

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The six-cycle and its complement have the Erdős-Hajnal property

Statement

The pair ${C_{6}, \overline{C_{6}}}$ has the Erdős-Hajnal property.

Facts & Assumptions

Given: The cycle $C_{6}$ .

[L1]

The pair ${P_{4}^{⋆}, \overline{P_{4}^{⋆}}}$ has the Erdős-Hajnal property (The star-expansion of the four-vertex path and its complement have the Erdős-Hajnal property).

[L2]

The star-expansion $P_{4}^{⋆}$ contains an induced $C_{6}$ (The star-expansion of the four-vertex path contains induced six- and seven-cycles).

[L3]

The Erdős-Hajnal property passes to hereditary subclasses (The Erdős–Hajnal property and each of its constants pass to hereditary subclasses).

Proof

technique · direct

1.1

Let $C$ be the class of graphs containing neither $C_{6}$ nor $\overline{C_{6}}$ as an induced subgraph. If $G \in C$ contained $P_{4}^{⋆}$ , then [L2] would force an induced $C_{6}$ in $G$ ; similarly, if $G$ contained $\overline{P_{4}^{⋆}}$ , then $\overline{G}$ would contain $P_{4}^{⋆}$ and hence $G$ would contain $\overline{C_{6}}$ . Thus every graph in $C$ is also ${P_{4}^{⋆}, \overline{P_{4}^{⋆}}}$ -free.

L1L2given

2.1

Therefore $C$ is a hereditary subclass of the class from [L1], so [L3] implies that $C$ has the Erdős-Hajnal property. This is exactly the statement that ${C_{6}, \overline{C_{6}}}$ has the Erdős-Hajnal property.

step 1.1L3∎

Depends on

Used by

Nothing in the library uses this result yet.

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

Maria Chudnovsky, Alex Scott, Paul Seymour, and Sophie Spirkl, Erdős-Hajnal for graphs with no 5-hole, Theorem 1.7 (standard reference, not scraped)
Shenwei Huang, Yiao Ju, and Yidong Zhou, Erdős-Hajnal beyond the five-vertex path, sentence after Theorem 1.9 (standard reference, not scraped)