ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-07

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 two induced co- $E$ witnesses behind the forbidden path runs

Example

In the ambient configuration of the preceding lemma, an induced $a - b - c$ with $u$ adjacent only to $a$ gives the six-vertex witness on ${x, y, u, a, b, c}$ . An induced $a - b - c - d$ with $u$ adjacent to $a, b, c$ only gives the six-vertex witness on ${y, u, a, b, c, d}$ . The other vertex $x$ of the ambient complete nonedge pair does not belong to this second witness.

Facts & Assumptions

Given: The two configurations in the Example.

[F1]

Co- $E$ is the complement of the stated $E$ graph (The $E$ -graph and co- $E$ ).

Proof

technique · direct

1.1

In the first configuration the nonedges are exactly $x y, y u, u c, c a, u b$ , the $E$ -edges in the order $(p_{1}, p_{2}, p_{3}, p_{4}, p_{5}, q) = (x, y, u, c, a, b)$ . In the second they are exactly $c a, a d, d u, u y, d b$ , the $E$ -edges in the order $(p_{1}, p_{2}, p_{3}, p_{4}, p_{5}, q) = (c, a, d, u, y, b)$ . Hence both graphs are co- $E$ .

F1algebra

2.1

Their displayed paths are induced and the mixed vertex has respectively two consecutive nonneighbours and three consecutive neighbours.

step 1.1

3.1

This verifies both finite witnesses.

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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

Huang, Ju, and Zhou, Erdős-Hajnal beyond the five-vertex path, Figure 8 (standard reference, not scraped)

The two induced co-EE witnesses behind the forbidden path runs