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.

A bipartite four-tooth comb has the co- $E$ structural partition

Example

Let $B_{1}, \dots, B_{4}$ be disjoint independent four-vertex sets. Add teeth $a_{i}$ , each complete exactly to $B_{i}$ , and add $v$ complete to every $B_{i}$ and anticomplete to all teeth; add no other edges.

Facts & Assumptions

Given: The displayed bipartite graph.

[F1]

Co- $E$ contains a triangle, on the images of $p_{1}, p_{3}, p_{5}$ (The $E$ -graph and co- $E$ ).

Proof

technique · direct

1.1

The graph is bipartite, with sides $⋃_{i} B_{i}$ and ${a_{1}, \dots, a_{4}, v}$ . By [F1] it is co- $E$ -free.

2.1

Choose $x_{i} \in B_{i}$ , set $X_{i} = {x_{i}}$ and $Y_{i} = B_{i} ∖ {x_{i}}$ . Each $Y_{i}$ is independent and co- $E$ -free.

step 1.1choose

3.1

Each singleton $X_{i}$ has a pure one-block blockade and one-vertex pattern, and every vertex in another block is anticomplete to it. Thus the structural clauses hold.

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

13 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, Lemma 6.4 (standard reference, not scraped)

A bipartite four-tooth comb has the co-EE structural partition