ExampleConstruction: AI-adaptedVerification: 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.

A rooted stable-tooth comb with two teeth

Example

Let $G$ be the graph on vertices

${v, a_{1}, a_{2}, b_{1}, b_{2}}$

with edge set

${v a_{1}, v a_{2}, a_{1} b_{1}, a_{2} b_{2}} .$

Then

$(v, ((a_{1}, {b_{1}}), (a_{2}, {b_{2}})))$

is a rooted stable-tooth comb in $G$ .

Facts & Assumptions

Given: The five-vertex graph in the Example.

[L1]

A rooted stable-tooth comb is a comb whose teeth form a stable set and whose root is adjacent to all teeth and anticomplete to all blocks (A rooted stable-tooth comb).

Verification

technique · direct

1.1

The two teeth $a_{1}, a_{2}$ are distinct, the blocks ${b_{1}}$ and ${b_{2}}$ are disjoint, each tooth is adjacent to its own block vertex, and neither tooth is adjacent to the other block. Thus $((a_{1}, {b_{1}}), (a_{2}, {b_{2}}))$ is a comb in $G$ .

given

2.1

The set ${a_{1}, a_{2}}$ is stable, the root $v$ is adjacent to both teeth, and $v$ is anticomplete to both blocks. Therefore [L1] identifies the displayed data as a rooted stable-tooth comb.

step 1.1L1∎

Depends on

A rooted stable-tooth comb

Used by

A cross-edge in a rooted stable-tooth comb creates an induced five-cycle Example

Dependency tree · two levels

3 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, Figure 1 pattern (standard reference, not scraped)