CorollaryStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-01‡ rests on unproved material (inherited)

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.

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Strong Perfect Graph Theorem, ‡ Substituting perfect graphs preserves perfection and ‡ Weak Perfect Graph Theorem. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

The bull graph has the Erdős-Hajnal property

Statement

The bull graph has the Erdős-Hajnal property.

Facts & Assumptions

Given: The bull graph.

[L1]

Every bull-free finite graph contains a clique or a stable set of size at least $∣ V (G) ∣^{1 / 4}$ , so bull-free graphs have Erdős-Hajnal constant $1 / 4$ (Every bull-free graph has a clique or stable set of size at least $∣ V (G) ∣^{1 / 4}$ ).

[L2]

A graph $H$ has the Erdős-Hajnal property exactly when the hereditary class of $H$ -free graphs has a positive Erdős-Hajnal constant (The Erdős–Hajnal property and an Erdős–Hajnal constant for a hereditary graph class).

Proof

technique · direct

1.1

By [L1], the hereditary class of bull-free graphs has the positive exponent $1 / 4$ .

2.1

By [L2], that is exactly the statement that the bull graph has the Erdős-Hajnal property.

step 1.1L2∎

Depends on

Used by

Dependency tree · two levels

9 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 2.3 (standard reference, not scraped)
Tung H. Nguyen, Notes on Recent Work on the Erdős-Hajnal Conjecture, Section 1 (standard reference, not scraped)