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 two six-vertex prime $H$ -graphs have the Erdős-Hajnal property

Statement

Both the left and the right six-vertex prime $H$ -graphs have the Erdős-Hajnal property.

Facts & Assumptions

Given: The left and right six-vertex prime $H$ -graphs.

[L1]

The bull graph has the Erdős-Hajnal property (The bull graph has the Erdős-Hajnal property).

[L2]

For a single graph, the Erdős-Hajnal property is equivalent to virality (For a single graph, the Erdős–Hajnal property, the polynomial Rödl property, and virality are equivalent).

[L3]

Deleting a leaf from each of two forbidden graphs preserves virality (Deleting a leaf from each of two forbidden graphs preserves virality).

[L4]

A graph and its complement have the same Erdős-Hajnal constants (A hereditary class has the Erdős–Hajnal property exactly when its complementary class does, with the same constants).

[F1]

In the left six-vertex prime $H$ -graph, deleting $ℓ_{1}$ or $ℓ_{2}$ leaves a bull: after deleting $ℓ_{1}$ , the triangle is $t_{1} t_{2} t_{3}$ with leaves $ℓ_{2}, ℓ_{3}$ , and after deleting $ℓ_{2}$ , the same triangle has leaves $ℓ_{1}, ℓ_{3}$ .

[F2]

The right six-vertex prime $H$ -graph is the complement of the left one by definition.

Proof

technique · direct

1.1

By [L1] and the direction $(1) \Rightarrow (3)$ in [L2], the singleton family consisting only of the bull graph is viral.

L1L2

2.1

Let $L$ be the left six-vertex prime $H$ -graph. By [F1], if we delete $ℓ_{1}$ from one copy of $L$ and $ℓ_{2}$ from another, both modified singleton families are the viral family ${bull}$ . Applying [L3] with the same graph $L$ in both leaf-deletion slots shows that the singleton family ${L}$ is viral. Using the direction $(3) \Rightarrow (1)$ in [L2], we conclude that $L$ has the Erdős-Hajnal property.

step 1.1L2L3F1

3.1

Let $R$ be the right six-vertex prime $H$ -graph. By [F2], we have $R = \overline{L}$ , so [L4] transfers the Erdős-Hajnal property from $L$ to $R$ .

step 2.1L4F2

4.1

Therefore both six-vertex prime $H$ -graphs have the Erdős-Hajnal property.

step 2.1step 3.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

19 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. IV. New graphs with the Erdős-Hajnal property, Theorem 1.4 (standard reference, not scraped)
Shenwei Huang, Yiao Ju, and Yidong Zhou, Erdős-Hajnal beyond the five-vertex path, Figure 2 discussion (standard reference, not scraped)

The two six-vertex prime HH-graphs have the Erdős-Hajnal property