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

Example

Start with the graph from A rooted stable-tooth comb with two teeth and add the extra edge $b_{1} b_{2}$ . Then the five vertices $v, a_{1}, b_{1}, b_{2}, a_{2}$ induce a copy of $C_{5}$ .

Facts & Assumptions

Given: The graph obtained from A rooted stable-tooth comb with two teeth by adding the edge $b_{1} b_{2}$ .

[L1]

In any rooted stable-tooth comb, a cross-edge between two different blocks forces an induced $C_{5}$ (A rooted stable-tooth comb with a cross-edge between two blocks contains an induced five-cycle).

Verification

technique · direct

1.1

The underlying five-vertex configuration is still a rooted stable-tooth comb with teeth $a_{1}, a_{2}$ , blocks ${b_{1}}, {b_{2}}$ , and root $v$ ; the new feature is exactly the cross-edge $b_{1} b_{2}$ between the two blocks.

given

2.1

Applying [L1] with $i = 1$ and $j = 2$ shows that the induced subgraph on ${v, a_{1}, b_{1}, b_{2}, a_{2}}$ is a copy of $C_{5}$ .

step 1.1L1∎

Depends on

Used by

A comb can have an edge between two blocks Counterexample

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, proof of Theorem 4.4 (standard reference, not scraped)