ExampleConstruction: Literature-sourcedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-16

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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.

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Complete and anticomplete disjoint pairs are $0$ -regular

Statement

If $X, Y$ are disjoint nonempty vertex sets that form a complete pair or an anticomplete pair, then $(X, Y)$ is $0$ -regular.

Facts & Assumptions

Given: A complete or anticomplete disjoint pair $(X, Y)$ .

[L1]

A pair is $0$ -regular when every pair of nonempty subsets $A \subseteq X$ , $B \subseteq Y$ has $d (A, B) = d (X, Y)$ ( $ϵ$ -regular pairs and self-regular vertex sets).

[L2]

In a complete pair all possible cross-edges are present, while in an anticomplete pair none are present (Edges between disjoint vertex sets; complete, anticomplete, pure and mixed pairs).

Verification

technique · direct

1.1

In the complete case, [L2] gives $d (X, Y) = 1$ , and every nonempty subpair $(A, B)$ also has density $1$ .

givenL2algebra

1.2

In the anticomplete case, [L2] gives $d (X, Y) = 0$ , and every nonempty subpair has density $0$ .

givenL2algebra

2.1

Thus the density difference is zero in either case, which is exactly the $0$ -regular convention in [L1].

step 1.1step 1.2L1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 6 results over 6 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, Definition 2.1.1 (standard reference, not scraped)

Complete and anticomplete disjoint pairs are 00-regular