CounterexampleConstruction: AI-generatedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-27

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.

If a tooth sees a foreign block, the structure is not a comb

Statement refuted

Refuted claim: a sequence of teeth and blocks is still a comb even if some tooth is adjacent to a foreign block.

Facts & Assumptions

Given: Two blocks $B_{1} = {b_{1}}$ and $B_{2} = {b_{2}}$ , and two teeth $a_{1}, a_{2}$ with edges $a_{1} b_{1}$ , $a_{1} b_{2}$ , and $a_{2} b_{2}$ .

Proof

technique · direct

1.1

The tooth $a_{1}$ is complete to its own block $B_{1}$ , but it is also adjacent to the foreign block $B_{2}$ .

given

2.1

The definition Combs in a graph requires every tooth $a_{i}$ to be anticomplete to every foreign block $B_{j}$ with $j \neq i$ . Step 1.1 violates that requirement, so this data is not a comb.

step 1.1given

3.1

Therefore the claim is false.

step 2.1∎

Depends on

Combs in a graph

Used by

Nothing in the library uses this result yet.

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

Shenwei Huang, Yiao Ju, and Yidong Zhou, Erdős-Hajnal beyond the five-vertex path, §2 (standard reference, not scraped)