DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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.

$ϵ$ -regular vertex partitions, equitable partitions, and refinement

Definition

Let $G$ be a nonempty graph and let $P = {V_{1}, \dots, V_{k}}$ be a partition of $V (G)$ into nonempty parts. For $ϵ > 0$ , the partition is $ϵ$ -regular if $\sum_{(i, j) : (V_{i}, V_{j}) is not ϵ -regular} ∣ V_{i} ∣ ∣ V_{j} ∣ \leq ϵ ∣ V (G) ∣^{2},$ where ordered pairs and the diagonal pairs $i = j$ are included ( $ϵ$ -regular pairs and self-regular vertex sets).

The partition is equitable if $∣ ∣ V_{i} ∣ - ∣ V_{j} ∣ ∣ \leq 1$ for all $i, j$ . A partition $Q$ refines $P$ if every part of $Q$ is contained in a part of $P$ . The empty family is the unique partition of the null graph and is equitable and $ϵ$ -regular by convention.

Depends on

$\epsilon$-regular pairs and self-regular vertex sets

Used by

Dependency tree · two levels

2 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

Y. Zhao, Graph Theory and Additive Combinatorics, Definition 2.1.7 and Theorem 2.1.20 (standard reference, not scraped)
D. Conlon and J. Fox, Graph removal lemmas, sec. 2.1 (standard reference, not scraped)

ϵϵ-regular vertex partitions, equitable partitions, and refinement