DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)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.

Combs in a graph

Definition

Let $ℓ \in N$ with $ℓ \geq 1$ , and let $w > 0$ . An $(ℓ, w)$ -comb in a graph $G$ is a sequence of pairs

$((a_{i}, B_{i}) : i \in [ℓ])$

satisfying the conditions below.

Here a vertex $a$ is complete to (respectively, anticomplete to) a set $B$ when the pair $({a}, B)$ is complete (respectively, anticomplete) in the sense of Edges between disjoint vertex sets; complete, anticomplete, pure and mixed pairs.

$(B_{1}, \dots, B_{ℓ})$ is an $(ℓ, w)$ -blockade;
the vertices $a_{1}, \dots, a_{ℓ}$ are distinct;
the set ${a_{1}, \dots, a_{ℓ}}$ is disjoint from every block $B_{i}$ ; and
for every $i \in [ℓ]$ , the vertex $a_{i}$ is complete to $B_{i}$ ; and
for all distinct $i, j \in [ℓ]$ , the vertex $a_{i}$ is anticomplete to $B_{j}$ .

The vertices $a_{i}$ are the teeth of the comb.

Depends on

Used by

If a tooth sees a foreign block, the structure is not a comb Counterexample
A three-tooth comb Example

Dependency tree · two levels

4 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 (standard reference, not scraped)