DefinitionDefinition: AI-adaptedProof: Not applicablejudge 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.

The $H_{5}$ -overlap-chain relation in one comb block

Definition

Fix a comb block $B_{i}$ . Let $X_{i}$ be the set of vertices of $B_{i}$ that lie in an induced $H_{5}$ . For $d, d^{'} \in X_{i}$ , write $d \sim_{H_{5}} d^{'}$ when there is a finite sequence $d = d_{1}, \dots, d_{m} = d^{'}$ in $X_{i}$ such that each consecutive pair $d_{r}, d_{r + 1}$ lies in one induced copy of $H_{5}$ contained in $B_{i}$ .

This is the $H_{5}$ -overlap-chain relation. Its equivalence classes are the $H_{5}$ -overlap classes of $B_{i}$ . The relation is reflexive (the length-one sequence), symmetric (reverse a chain), and transitive (concatenate chains).

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, and Zhou, Erdős-Hajnal beyond the five-vertex path, proof of Lemma 6.4 (standard reference, not scraped)

The H5H5​-overlap-chain relation in one comb block