LemmaStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-16

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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.

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A small energy increment makes fine-pair densities close to their coarse densities almost everywhere

Statement

Let $Q$ refine a vertex partition $P$ of an $n$ -vertex graph. For $x, y \in V (G)$ , let $P_{x}, P_{y}$ be their $P$ -parts and $Q_{x}, Q_{y}$ their $Q$ -parts. If $q (Q) - q (P) \leq ϵ^{3},$ then fewer than or equal to $ϵ n^{2}$ ordered pairs $(x, y)$ satisfy $∣ d (Q_{x}, Q_{y}) - d (P_{x}, P_{y}) ∣ > ϵ .$

Facts & Assumptions

Given: Partitions and a parameter satisfying the Statement.

[L1]

Energy is the weighted mean square of part-pair densities, and refinement cannot lower it (The mean-square density, or energy, of a vertex partition, Energy lies in $[0, 1]$ and cannot decrease under refinement).

[L2]

For a nonnegative finite random variable $X$ and $a > 0$ , Markov's inequality gives $\Pr (X \geq a) \leq E X / a$ (Markov's inequality on a finite probability space).

Proof

technique · direct

1.1

Choose an ordered vertex pair $(x, y)$ uniformly and set $Z_{P} = d (P_{x}, P_{y})$ and $Z_{Q} = d (Q_{x}, Q_{y})$ . On each coarse pair, double-counting gives $E (Z_{Q} ∣ P_{x}, P_{y}) = Z_{P}$ .

givenL1algebra

2.1

Expanding the square and using the conditional identity in step 1.1 gives $E ((Z_{Q} - Z_{P})^{2}) = E (Z_{Q}^{2}) - E (Z_{P}^{2}) = q (Q) - q (P) \leq ϵ^{3} .$

step 1.1L1algebra

3.1

Apply [L2] to $(Z_{Q} - Z_{P})^{2}$ with threshold $ϵ^{2}$ . The probability that $∣ Z_{Q} - Z_{P} ∣ > ϵ$ is at most $ϵ^{3} / ϵ^{2} = ϵ$ .

step 2.1L2algebra

4.1

Multiplying this probability by the $n^{2}$ ordered vertex pairs proves the asserted exceptional-pair bound.

step 3.1algebra∎

Depends on

Used by

Strong regularity with linearly large representative subsets and no irregular representative pair Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 18 results over 7 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

Y. Zhao, Graph Theory and Additive Combinatorics, Lemma 2.8.7 and Exercise 2.8.8 (standard reference, not scraped)