ExampleConstruction: Literature-sourcedVerification: AI-generatedprecheck 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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

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

1.1givenL2algebra

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

1.2givenL2algebra

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

2.1step 1.1step 1.2L1∎

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

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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 2.1.1 (standard reference, not scraped)

Complete and anticomplete disjoint pairs are 00-regular