LemmaStatement: AI-adaptedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-26

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.

An $ϵ$ -regular pair is $ϵ^{'}$ -regular for every $ϵ^{'} \geq ϵ$ with $ϵ^{'} > 0$

Statement

Let $0 < ϵ \leq ϵ^{'}$ . If $(X, Y)$ is an $ϵ$ -regular pair, then it is an $ϵ^{'}$ -regular pair.

Facts & Assumptions

Given: Reals $0 < ϵ \leq ϵ^{'}$ , disjoint nonempty vertex sets $X, Y$ , and an $ϵ$ -regular pair $(X, Y)$ .

[L1]

An $ϵ$ -regular pair requires the density deviation bound for all subsets $X^{'} \subseteq X$ , $Y^{'} \subseteq Y$ with $∣ X^{'} ∣ \geq ϵ ∣ X ∣$ and $∣ Y^{'} ∣ \geq ϵ ∣ Y ∣$ ( $ϵ$ -regular pairs and self-regular vertex sets, Edge counts and densities between nonempty vertex sets).

Proof

technique · direct

1.1

If $∣ X^{'} ∣ \geq ϵ^{'} ∣ X ∣$ and $∣ Y^{'} ∣ \geq ϵ^{'} ∣ Y ∣$ , then also $∣ X^{'} ∣ \geq ϵ ∣ X ∣$ and $∣ Y^{'} ∣ \geq ϵ ∣ Y ∣$ because $ϵ^{'} \geq ϵ$ .

L1algebra

2.1

The $ϵ$ -regularity of $(X, Y)$ therefore gives $∣ d (X^{'}, Y^{'}) - d (X, Y) ∣ \leq ϵ \leq ϵ^{'}$ . This is exactly the definition of $ϵ^{'}$ -regularity.

step 1.1L1algebra∎

Depends on

Used by

Nikiforov: for every H and every ε∈(0,1/2) there is δ>0 such that every graph G with ind_H(G)<(δ|V(G)|)^|V(H)| has an ε-restricted vertex set of size at least δ|V(G)| Theorem

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

Y. Zhao, Graph Theory and Additive Combinatorics, Remark 2.1.3 (standard reference, not scraped)

An ϵϵ-regular pair is ϵ′ϵ′-regular for every ϵ′≥ϵϵ′≥ϵ with ϵ′>0ϵ′>0