DefinitionDefinition: AI-adaptedProof: Not applicablejudge 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

Definition

Let $G$ be a finite graph. A rooted stable-tooth comb in $G$ is data

$(v, ((a_{i}, B_{i}) : 1 \leq i \leq t))$

with $t \geq 1$ such that:

$((a_{i}, B_{i}) : 1 \leq i \leq t)$ is a $(t, w)$ -comb in $G$ for some $w > 0$ , in the sense of Combs in a graph;
$v \notin {a_{1}, \dots, a_{t}} \cup B_{1} \cup \dots \cup B_{t}$ ;
the set ${a_{1}, \dots, a_{t}}$ is stable; and
the root vertex $v$ is complete to ${a_{1}, \dots, a_{t}}$ and anticomplete to $B_{1} \cup \dots \cup B_{t}$ .

The word "stable-tooth" records condition 3: the teeth form a stable set. Condition 2 makes the disjoint-set predicates in condition 4 well defined in the sense of Edges between disjoint vertex sets; complete, anticomplete, pure and mixed pairs.

Depends on

Used by

Dependency tree · two levels

5 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, Theorem 3.1 (standard reference, not scraped)