DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generatedjudge 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.

E overlap chains inside one comb block

Definition

Fix a block $B_{i}$ of a finite graph comb (Combs in a graph). Let $E_{i}$ consist of all six-vertex subsets of $B_{i}$ inducing the graph in The $E$ -graph and co- $E$ . Put $X_{i} = ⋃_{S \in E_{i}} S$ and $Y_{i} = B_{i} ∖ X_{i}$ . For $d, e \in X_{i}$ , define $d R_{i} e$ if there exist $m \geq 0$ and vertices $d = d_{0}, d_{1}, \dots, d_{m} = e$ in $X_{i}$ such that each consecutive pair is contained in some $S \in E_{i}$ . A zero-length chain is allowed. If $E_{i}$ is empty then $X_{i}$ and the relation are empty. We call this the $E$ overlap chain relation.

Throughout this definition, “the graph in The $E$ -graph and co- $E$ ” means the $E$ -graph defined there, not the co- $E$ graph.

Depends on

Used by

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

Huang–Ju–Zhou, Erdős–Hajnal beyond the five-vertex path, §6.2, Lemma 6.5, overlap relation (standard reference, not scraped)